I would think that the inertia of the arrow has to be taken into account when discussing cross-wind effects. FL mentioned that once the object in question is embodied in fluid or air- it will move at the speed of the surrounding fluid or air.

BUT, our arrow is launched from a static position in relation to the crosswind. Besides the amount of drag on the arrow, the inertia of the arrow will also play a part. Kind of like dropping a cork and a half-full 55 gallon drum off of a bridge into a river. The cork will be brought up to speed with the river current faster than the drum due to the inertia of the drum resisting the initial flow.

Maybe this effect of inertia is negligible in arrow weight comparisons and crosswinds, but I think it should still be considered?