Would pyrokinesis be an effective defense?

I propose another theory in which to 'stop' the bullets with flames or pyrokinesis. Creating a wall of flame is pointless, as other before me have said, it would need to be very hot and very thick to vaporize a bullet.

So instead, create a vacume of air and oxygen within a circle of flames. If possible the barrier should stop the bullets momentum or drag it down lower. And if your pyrotechnic is skilled enough, they should be able to control the vacume of air and fold the flame within to change or divert the bullets.

As my Uncle once said, "Not everything can be solved with fire, but a damn lot can."

Other answers have focused on the sheer amount of heat needed to vaporize the bullet, but I think (I don't really know -- ask someone who's good at physics) that your first obstacle to get over would be air displacement. It's worth considering that the vortex left in the wake of the bullet's path may effectively extinguish any flame nearby due to starvation.

Depending on your pyrokinetic's origin of ability (e.g. on their person v. at a remote location in their field of vision) combined with their level of detail and speed, the more effective defense (other than obscuring themselves with a smokescreen/the sheet of flames or distracting/frightening the attacker) may be to preempt the shot and cause an early ignition, although this would require considerable attention to detail so that the projectile is not aimed somewhere equally damaging.

Air temperature might also be taken into account -- is there any property of air buffering or flight physics that can be altered by a drastic change in temperature?

The pyrokinetic does not need to destroy the bullet to not get hit.

The pyrokinetic could blow the bullet off course.

Flame is simply air so hot that it is incandescent. When you heat a gas it expands in volume. The faster you heat it, the faster it expands. Rapidly expanding gas is an explosion.

https://www.physicsforums.com/threads/how-much-does-air-expand-with-heat.267530/ PV = nRT; V is volume and T is temperature. So if pressure is stable an increase in temperature causes the same increase in volume. A candle flame is 900C. From 30C to 900C means a volume increase of 30x. An oxyacetylene torch is 3000C; from 30C to 3000C is 100x the volume. If you turn 1 cubic meter of gas into 100 cubic meters of gas in 1 second, that would make a 360 kph wind or 223 mph wind.

A sudden expansion of heated gas (flame!) in the path of a bullet could blow it off course. There is no question that wind can push a bullet off course. The deviation has to do with the mass of the bullet, the speed of the wind, and the flight time of the bullet (or how long the wind gets to push it). There are published tables for shooters. Here is one.

Multiplying the values for 20mph wind by 10, a 200mph wind could push a bullet this size 16 inches over 100 yards and farther over longer distances.

This would not always save the target. The deviation of the bullet would depend on the volume of gas heated, how hot it got, the speed and mass of the bullet and the distance of the shot. The pyrokinetic should practice. But a bullet does not need to miss by much to miss.

Your pyrokinetic will be lucky if his assailant is only using lead bullets. Most ammo today is steel jacketed and there are much more durable materials being used to make armor piercing projectiles.

So, with that in mind, mayhaps we should look for a means of using your pyrokinetic's talents in more imaginative ways. And, as always, the devil is in the details.

For instance, if his assailant happens to be a sniper then your pyrokinetic may never even see it coming, so mounting any kind of defense would become a moot issue. So we must assume you're looking for a way to preserve the life of your pyrokinetic in situations where he has some prior warning of impending doom. Thus he has time to mount a formidable offense and should do so prior to his assailant firing his weapon. In this scenario your pyrokinetic can use his amazing talents to fry his arch nemesis to a crispy thin strip of bacon.

On the other hand, if your pyrokinetic hasn't the stomach for such violence then he could just warm his assailants weapon to a point where holding on to it long enough to take aim is out of the question. That always produces opportunities to lighten the mood for all observers when they see the assailant dancing around trying to find a way to cool his trigger finger.

Now, let us assume the weapon has already been dis-charged in your pyrokinetic's direction. As others have pointed out these projectiles are not walking. If your pyrokinetic hasn't already got a plan ready and begun its initiation, it isn't likely he is going to survive long enough to figure out what to do about the bullet, except maybe figuring out how to get it out from under his skin.

Likewise, if he knows the weapon is about to be discharged in his direction and he has time to flame on, simply trying to generate enough heat to melt the bullet isn't going to be the safest method of ensuring his survival. So let's assume your pyrokinetic has very precise control over his amazing ability and has had enough practice to use it effectively in concentrated bursts. As opposed to a wall of flame, your pyrokinetic should discharge an extremely intense beam of fire about twice the size of a standard bullet in circumference and thrust it in a trajectory that will ensure collision with the bullet at a force much greater than any bullet, irrespective of powder pack or material used, could resist. In this way he simply slows the bullet down very quickly until it drops at his feet, still smoking hot and glowing a bright orange/red, capping off the scene with a loud hiss and sizzle as the ground it has fallen on is damp.

Does that work for you?

You need to calculate Heat capacity of air and Heat capacity of lead going through it. Boiling point of lead is 1750 °C. So it is 1730 difference to make. Lets assume that bullet will only gain heat from Air it went through - so basically it has same volume (or very similar).
Air has higher mass heat capacity, but it depends on mass, not volume.
So we take volume heat capacity because of same volume of air and bullet. That is 0.00121 for Air and 1.44 for Lead.
Lead/Air*1730 = 2 058 842°C/K or 3 705 947 degrees Fahrenheit.

You shouldn't care about velocity, unless you think "only 10% of heat would be absorbed by bullet" - in that case you will need 10x higher temperature.
"Small lump of metal flying at you" - Unless you have 2000+ degrees around you, it will change back to metal. But more likely it would stay as hot steam for few seconds - just don't breathe it.

What is pyrokinesis actually?

Is it a gathering of molecules from the surroundings (air, dust, etc) and then superheating and accelerating them?

If your hero can throw a fireball, could he also generate a mini-fireball that travels at similar speeds as a bullet? Could he generate a fire bullet fast enough and at the correct mass+velocity to counter the energy of the bullet?

You could stop the forward momentum, but the bullet would probably shatter and you would have fragments spreading out from the point of impact. If the fire bullet was a larger mass, then it could envelop the bullet and pretty much vaporize the fragments. That fire bullet would also probably "shatter" and dissipate back into the air after impact.

I'm in the group that think of pyrokinesis as a specialized form of telekinesis where the pyrokinetic energizes molecules to induce heat and ultimately fire. Take this far enough and I don't see why your pyrokinetic couldn't end up generating plasma. CERN has produced a type of plasma that reaches 5.5 TRILLION degrees celcius, which exceeds the above temperatures about 2 million times over. Your pyrokinetic could also have some degree of electro-magnetic influence as well, which would be a bit of a requirement for controlling plasma.

As user @rytan451 states, it's not really feasible to flat out vaporize the bullet (though I'd be curious to see the calculations he used to arrive at 3ish Mk). What you could do is super heat the air next to the bullet and use the expansion of the gas to deflect the trajectory slightly. For this to be effective however, your pyrokenetic would have to have superhuman reflexes and precision in order to intercept the bullet as it travels.

For another trick in using pyrokinesis to block bullets:

Depending on how long the sheet of fire can be maintained for, it could be used simply to obscure the view of the shooter.

It's pretty difficult to see through fire, especially with the addition of smoke, which in an enclosed space would cause the shooter's eyes to suffer as well (though that might also happen to your pyromancer - I would recommend goggles).

Alternatively, depending on how well the heat can be manipulated, it could be possible to conjure a mirage, in which heat bends light rays and creates a different picture than what is really there, which could be used to distort what the shooter sees and cause them to miss.

Whilst these techniques don't necessarily stop bullets in the air, it can stop them from being fired accurately, which is just as useful.

It would be very effective if you assume one thing:
Your pyrokineter is actually electro/air (apart from Firefly from DC pyros don't use any kind of fuel to burn and/or flamethrowers).
Mostly because it's easier to use nitrogen from the air and add a little of electricity to create plasma.
Plasma has the advantage that it can:

1. cut through metal
2. plasma can be magnetically charged and would push away Pb

So you would not only have to deal with smaller particles, but it would be easier for the field to push them in different directions.

Think of it as a bush that changes the trajectory of a bullet.

Pyrokinetic "sheet of fire" can not be fire at all. What it has to be is a region of space where thermal molecular velocity is increased by the kinetic's power, to the point where oxygen-nitrogen recombination occurs (which means that pyrokinesis is not environmentally friendly, as nitrogen oxides are responsible for acid rains).

The same effect is a defense against most kinds of projectiles: their macroscopic kinetic energy is neutralized, they can be heated or fractured until they disintegrate, and even the shock wave of an explosion - being kinetic energy of air molecules - can be diverted.

The possible limits are where does this energy go - we can somehow transmit it without physical contact, but it's best if we don't repeal the laws of thermodynamics - and how much of it can be transferred by the kinetic, and for how long. Also, how long does it take for the kinetic power to "lock" on the incoming bullet.

Since the power for a sheet of fire is above what can be generated by a human body, the energy must be supplied by something else (alien nanomachines, trans- dimensional demons...). Maybe that something else can take care of monitoring airspace and target acquisition.