Global Entry has nothing to do with the airline you fly. It is simply a way to get through US customs and immigration checks quicker (sometimes an hour quicker! depending on your arrival airport) on your return to the US by air.
TSA Pre Check, which you automatically receive if you get accepted into Global Entry, does depend on which airline you fly with the airline's participation required to get the TSA Pre indicator on your boarding pass. This gets you access to the Pre Check lanes at US airports where it is available. If you fly a lot, this can add up to a lot of time savings.
Pre check only works on flights departing from the US, Global Entry only works for flights arriving in the US. So no benefit to you if you connect through airports outside the US.