Just carried out a little bit of original research and found that the projectile behavior of a crossbow bolt out of the crossbow is unnafected by the velocity of the player when travelling by elytra. Oddly enough, the velocity of lingering potions does seem to be affected by the player's elytra velocity: When elytra flying at a high speed, the splash potion seems to fly further when thrown while stationary. Further research will examine the behavior of projectiles during various transportation methods, and exame different projectiles, like ender pearls, snowballs, arrows fired from a bow (as opposed to a crossbow), and chicken eggs.
Personally, I feel that the crossbow bolt's velocity should change if the player is travelling at speed with an elytra. While it might make the crossbow a little more difficult to aim, it would feel more natural and give the player the advantage of increased projectile speed, and therefore range, if firing forwards.