I think you’re conflating two different questions here. These questions are really focused on a central question of “How would the internet work with latency measured in minutes, when most systems are configured around latency in milliseconds?” And the answer there is “We’d have to change how some systems work, and others wouldn’t be feasible.” Barring some method of FTL communication (which would be an instant Nobel prize in physics), you’re never going to get real time instant messaging between Earth and Mars, but async methods of communication will work fine, albeit with more latency. But we’re able to exchange digital data across planets now, it’s just that the public internet is built around the assumption that the speed of light is only going to account for ~100ms of lag.

If you take an assumption that high latency digital communication will be feasible relatively soon after we have people living off planet (which is a reasonable assumption), networks like the fediverse will function with a lot of caching, as tikitaki mentioned. You’d never have perfect sync, but the biggest challenge would likely be how much bandwidth is required to keep different caches in sync.

To answer your specific question, you’d probably start an initial cache on planet, then keep it in sync during transit.