Is there a way to move a grabbed creature with you? I mean dragging or carrying the grabbed creature with you.
Not currently as written, though I don't see why making it an athletics check with the same rules and distances as shove would be a problem.
I'm assuming an actively hostile creature, kicking and screaming?
Because if it, say, an unconscious (or not) ally, use the carrying and bulk rules.
Because those are clearer when it comes to the bulk of characters and their total equipment? /s
It'll often be the case that mechanically it's easier to Shove the body along than carry or even drag it. The bulk system breaks down profoundly when discussing anything other than standard gear on a single entity.
The Roc entry in the Monster Manual has a special ability "Snatch" to allow it to do fly off with a grabbed crature. So likely not intended to be generally possible without a similar ability.