The relationship between flow and driving pressure depends on the flow profile: is gas flow laminar (all gas molecules move orderly at the same time in the same direction) or turbulent (there is no orderly flow as above, and there are many eddies)? We assume that you are not interested in any details. So please accept that, taking into account these extremes and by applying Poiseuille's and Darcy's equations, as appropriate, it can be shown that in the case of isometric growth flow should increase as a power function of volume, where the power may vary between 0,67 and 1,0.
PS: If we take into acount that lung elastic recoil may play a role - it takes too much time to explain quickly - then the power k may be as large as 1.37. It also matters whether the pressure loss in airways should only be attributed to acceleration of gas molecules or also to frictional pressure losses, but this is similarly specialist matter.