Rush Teleporting on Damage

In Mega Man 4 (as well as some ROM Hacks of it), Rush will teleport under Mega Man when he takes damage. This is actively used in the run as a strat and a race-backup.

Rush Coil in Pharaoh Balloon Jump

By having (unused) Rush Coil on screen, if you take damage in the balloon jump from the bee, Rush will "save" Mega Man.

Wily 2 Jet Boost

On the last long screen before the boss door of Wily 2, you can take a damage boost to pull Rush Jet along faster.