Each airline will run their own background check.

Leaving an airline as a new-hire might well blacklist you from ever getting called by that airline again. Depends on the airline and their needs in the future. And also your reasons... if you resigned to care for an ill family member and had no other option, they'd probably take you back later.

It would show on your record but if you had minimal air gap between leaving and starting at the next airline it would be obvious that you just got a better (for you) offer. As opposed to fired for some reason. They know that if you had a two week employment gap, that you already had the next job lined up because you can't apply, interview, and get a legacy class date in two weeks.

Since we're talking legacies, ie career destination jobs, I'd be somewhat less concerned about burning bridges. If you get on with a legacy which domicile geography that works for you, you're probably not leaving until you retire.