After a pillager tries to shoot a player from a distance, they'll slowly walk for a few seconds and then chases the player. It didn't happen in 1.15.2 as they would run towards the player after shooting.
Steps to Reproduce:
Spawn a pillager
When it starts charging it's cross bow, walk backwards away from it
Let the pillager shoot, then start walking
Hi there!
I can confirm for release 1.16.2.