If I were going to design an original game / campaign around this premise, here’s what I’d do. Time travel is consciousness-only – your body doesn’t actually pass through time. You are psychically linked to the other PCs, so you can travel to any time when any PC is alive in their own body. When you do, you possess the body of someone who is connected to that PC. At character creation, you each get to identify some number of things about your character’s life situation that are problems for you and you want to change. You and your time travel crew can change those things, but whenever you are offered “succeed at a cost” in the time period when you are playing, the cost takes place in another PC’s home time period – and you don’t know which one until you travel there and find out. Additionally, the ways in which your relationships with the other PCs change over time affect the ways that your host bodies/identities behave. So for example if you have a huge fight with a PC in one time period, you might travel to another time period to find out that your host character is not speaking to that person’s host, or that they have a reputation for being incredibly short-tempered, or some other way that that relationship change resonates. The goal of the campaign is to have all players happy with the current version of the timestream at the same time.