This is the same issue as MC-153172, which hasn't been fixed, despite the parent issue being marked as such.

How to reproduce:

1. /effect give @s minecraft:jump_boost infinite 9 true

2. Shoot arrows from a crossbow while jumping
3. Test the same thing with a bow for comparison

Expected result

The player's movement would influence the trajectory of projectiles fired from crossbows.

Observed result

The player's movement does not affect the trajectory of projectiles fired from crossbows. I believe it behaves the same when a pillager uses the crossbow.