[S/M] Egg Move Breeding Question

[CptObvious]

I'm slowly starting to figure out breeding. But if I have a pokemon with a learned egg Move and I want to continue to breed that Pokemon that has the egg move, does the egg move pass down or is the only way to keep breeding the parents to keep getting the Egg move.

Ex. I bred Smeargle and Bewear and got Stufful with Ice Punch. If I want to pass Ice Punch down can I evolve that Stufful to Bewear and breed to keep ice Punch? Or is the only way through Smeargle and Bewear?