If you don't want to violate causality then you can only have a few types of faster than light travel.
One is to not allow it to slow down. Another is to only allow it to go in one direction and hope the universe is infinite.
Otherwise it is rather trivial to use regular slower than light motion between some FTL trips to generate a time machine.
As for what kind of FTL spacetime allows, generally you need some exotic matter, which means negative energy density at the very least.
Now if you are contrasting this with space expansion that makes you get places slower. Sometimes people describe that as making things far away from you move away faster than light but it doesn't allow you to get places faster if the space between you expands.
If the universe contracted into a big crunch that would be more like an FTL. But that has the same kind of problems with slowing down and not having choices about which directions to go in. That makes it hard to build a time machine.
There are multiple ways to contract and expand space, there are even FTL metrics that don't contract the space between you and your destination they compress it in one direction (the direction you want to go) and spread it out in the other so the space doesn't become less between you.
And the kind of contraction that a Big Crunch has when the density of an isotropic homogeneous universe exceeds the critical density is a different kind of contraction than the warp drives have (they have one related to exotic matter).
So you can have FTL without contracting space and you can have contracting space without FTL.
And you can have time travel without a warp drive or a wormhole or any curvature at all, so it's hard to impose the causality condition you want. It's even hard to say when something is FTL when locally things are moving less than c.
Your question might be too broad. But to connect it together some people think you can't force a time machine to develop without exotic matter and thus we don't have to worry about causality and the warp drives and then we just have the regular space contraction. Which isn't really FTL even if the distance between the two objects decreases at faster than light.