I think so far the campaigns are designed to be mostly stand alone, with only vague references to the previous ones, e.g. an occasional returning NPC or something[1]. Or the second one was designed that way, anyway, at least from what I understand (having not yet watched it myself, of course), and I seriously doubt the third one will be any more tied to the previous ones beyond that, either.

[1] - Like how Allura and Kima apparently make appearances in Campaign 2. Since the third one is apparently going to be set in Marquet, I would personally expect at least a cameo appearance from J'mon Sa Ord and possibly Gilmore (or at least maybe his parents or his husband anyway).