First, I intend to be able to define simple "dependencies", but IMO that's not the interesting part.
I am going to use rw mutexes on every file with staged changes. If 2 steps are running against disjoint files there is no problem. If 2 steps are running against "check" (not "fix" which edit files by convention), they can also run at the same time. The only scenario where blocking is necessary is when you have 2 "fix" or 1 "check" and 1 "fix" on an intersection of their files.
For that scenario there is going to be a setting that you can enable to run a steps "check" command first (only if this scenario is about to happen, if no files will be in contention it will simply run "fix"), and if "check" fails, then it will switch to "fix".
This is my current idea and in my head I think it would work. There are caveats, notably running "check" and then "fix" might be slower than just waiting and running "fix" once which is why it needs to be optional behavior. You also may not care about things stomping on each other (maybe the linters themselves have their own locking logic).