How much air movement is need to produce a given thrust force?

It's not air that it is being pushed, it is the exhaust gases.

There are two ways to compute the thrust:

1. Momentum conservation: the rate at which momentum is given to the exhaust gas must be equal and oposite the force on the rocket by Newton's third law. As a result $$ F_{rm thrust} = dot m v_{rm exhaust}, $$ where $dot m$, the rate a which fuel is being converted to hot high-pressure exhaust gas, has units of ${rm kg.s}^{-1}$. Then ${rm kg s}^{-1}$ times m s$^{-1}$ gives units of Newtons.

2. Force balance: The force on the rocket is due to the fact gas pressure in the combustion chamber is pushing on all the walls of the chamber, but not on the area of the hole through which the exhaust gas is escaping. In the hole, the pressure is acting only on the gas, which is not attached to the rocket. As a consequence
   $$ F_{rm thrust}=c P_{rm chamber} A_{rm nozzle} $$ where $P_{rm chamber}$ is the pressure of the hot gas in the combustion chamber and $A_{rm nozzle}$ is the area of the hole (i.e the throat of the nozzle) and $c>1$ is a correction due the fact that there is also unbalanced gas pressure on the walls of the expanding part of the nozzle. The gas continues to accelerate in the expansion part of the nozzle until it reaches $v_{rm exhaust}$ as it leaves the nozzle and no longer exerts a force on the rocket.