What's the proper science-driven way to treat mild scrapes/abrasions (that have a bit of dirt in them) from a mountain biking crash?

The problem you are having is probably the difference between First Aid and medical treatement. From Wikipedia...

First aid is the first and immediate assistance given to any person suffering from either a minor or serious illness or injury, with care provided to preserve life, prevent the condition from worsening, or to promote recovery

Giving tetanus shots, cleaning wounds and even applying disinfectant/antibacterials is not first aid. Giving pain releif is usualy not first aid, but common. When medical treatement facities are a long way off First Aid takes treatement further - how you treat someone where an Ambulance is called and on the way may be very different to how you treat someone 3 days ride from the nearest road.

How should you treat a MTB abbrasion - Cover with dressing or bandage and then visit a medical center for treatement as soon as practical. Strictly speaking, advice on how to treate a wound is Medical Advice and Off topic for this site. One big problem is every wound is very different and what I call a large complex wound, you may think is trivial. The locatio of teh injury, the envroment where it occured greatly changes the risk factors for infections and disease. If I give advice, I am unlikely to consdier diseases that might occur in a tropical rain forest, and while I might say "if its large, vist a doctor" what I think of as large might be very different to yourself.

Best thing you could do is attend an outdoor, ideally MTB specific, first aid course.

Briefly, a quick rinse with clean water, and possibly dressing it, are probably all you need to do on the trail. This assumes a day visit to a typical trail centre or cross country riding near to facilities in a country with a reasonably accessible healthcare system, i.e. most mountain-biking. This was also confirmed by an outdoor first aid course I did recently (these are a very good idea).

You don't know what's in the dirt, so studies discussing antibacterial properties of certain mineral compounds aren't very relevant, even if they're effective against the bacteria that might be an issue, in the wild as opposed to in the lab. As an extreme example, consider coming off into fresh dung on a rock - slippery, unhealthy, but still capable of causing a graze.

Alcohol wipes are now deprecated for wound cleaning; first aid kits now include sterile saline wipes instead, but apart from using a folded corner to flick particles of dirt away, they're not very effective.

Rinsing with your water bottle is about as good as you'll get for cleaning while on the trail. This is a good reason to have plain water in (one of) your bottle(s) - most other drinks would not be such a good idea (similarly, washing muck out of important bike parts). Getting enough flow from a Camelbak isn't so easy, and you need to have plenty of water with you to do this. Some ride leaders I know carry a spare bottle that they don't drink from, for this, dehydrated novices etc., but in general the wounded person's own bottle should be first choice.

Once the wound is fairly clean it may be a good idea to dress it to keep it that way, but on something like a knee that flexes a lot while riding, this can be more trouble than it's worth, though if you're bleeding probably it's a good idea to cover the wound. Large stretch fabric sticking plasters (band aids) are ideal, though in my riding first aid kit I have a selection of these and bigger dressings. I carry more than most people as I'm used to leading kayaking and road riding. A more thorough clean might be needed when you can, e.g. a shower, or several bottles of water when there's a tap.

After a recent tumble leading to unsightly grazes on my knee and elbow (and that first aid course) my preferred dressing for wounds on moving parts is now the sort with a sterile pad and built in conforming bandage - it doesn't rely on adhesion so is more sweat-proof, and flexes well, while looking rather like a cartoon bandage. Zinc oxide tape is a also good for holding down dressing, if you use plenty.

Keeping your vaccinations up to date (tetanus being the most relevant) is an important general rule, regardless of what you do at the time of the injury.

Simple skin abrasions/grazing:

Symptoms: bleeding, can still walk.

• Sit the patient down out of the way. Let the blood flow for a short time to help push out embedded debris. Rinse the area with clean water, and apply some cloth and pressure. Ideally raise the injured area above the level of the heart to decrease pressure and let the wound clot. Give the patient liquids, talk calmly. They might appreciate a Gel or some food.
• After 5 minutes wet the cloth and peel it off. Ideally bleeding will have stopped. Check the bike over, and ride on home when ready.

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^{The rest of this got a bit OTT but left for completeness.}

Impact

Symptoms: Hurts, but no/minimal bleeding. It is a bruise that hasn't coloured-up yet.

• Sit and relax the patient. Apply anything you have that is cold or cool like water. If its an ankle, remove the shoe. Get patient to flex the muscle to stop it stiffening up. Head straight home, the injury will swell up. When home, get ice on it immediately.

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Broken bone

Symptoms: limbs at wrong angles, can't use limb.

• Call ambulance. Calm patient, give water, keep them warm, etc. You will need to make a judgement decision if getting the patient out is possible - can they walk with arm in a sling? If a leg then they're not walking or riding, so assign someone in the party to meet the ambulance, who will need a stretcher. If you're well in the backcountry, it might be worth looking to make up some way to carry the patient, though this is not easy.

Joint separation

• I'm not familiar enough with this one. Probably similar to broken bone - the patient might be able to walk but not ride. [TBC]
• An injured person may be able to sit on their bike and steer but not pedal, so you can use an inner tube as a towrope. Slow and steady.

Head/neck/spine injury

• Call the ambulance. Similar to broken bone - you're not walking out. Keep the patient warm and comfortable. Keep them awake too - don't let them fall asleep. Call the ambulance.

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If you've gone far enough off-road, an ambulance might not be possible and a helicopter may be required.

If you're out of cell phone coverage, assign a party member to ride to somewhere with coverage and relay details to emergency services. That may be up a hill as the quickest solution.

Have no hesitation in requisitioning help. It is totally OK to flag down others - that 4WD fishing in the distance may be well equipped, or that farmer over the valley may have radios that work where your cellphone doesn't.

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My "immediate" responses in event of an accident:

1. Get it safe - that means getting off the track/road. If the injury is bad enough that moving might be unsafe, then stop oncoming road or trail traffic - send someone with a torch/flashlight if possible.

2. Identify the injured - if there's more than one down then you have to quickly prioritise. Ignore anyone whose walking around complaining about scratches on their carbon, focus on anyone who is not moving.

3. Assess the injuries - talk calmly. Evaluate if its grazing and wind knocked out, or something worse. Apply first aid as required.

4. Identify your resources - how many able bodied people are there, including bystanders and anyone who has stopped to help. Assign a person to each task that needs doing:

   • Sit with each injured person and administer first aid as required.
   • Clear the road of bikes/debris
   • Take some photos for evidence, (like licence plates)
   • Contacting your nation's emergency services line 911, 111, 999, 000, or whichever.

All of this happens really quickly, and that list is a bit stilted and formalised. In reality its a lot more parallelised.

If you think you need it, call for an ambulance sooner than later. They generally take a while to get anywhere. Even having your national emergency service on a phone can be helpful guidance.

If you're riding on your own, well you'll have to do it all yourself. Ideally, don't go too bush alone, or if you do then go prepared. An EPIRB is a reasonable addition to your kit if you're in the backblocks.