There are two separate issues here.

1. Whether the carriers interline. If they interline, then the question is whether carrier #1 will interline across tickets and under what circumstances. If they don't interline, the entire issue is irrelevant.

2. Whether there is a bag drop for the departing carrier at the arrival location. This matters in the US because Customs is at the first POE. Thus, interlined or not, one still has to pick up the bags and drop them somewhere. Even if tagged.