As per the title. I have 2 shimano mechs and both of them act the same way on my hardtail. On the smallest cog it takes 3 or so clicks before it moves up & will typically jump 2 cogs before going all the way. It’ll run out of clicks before going into the biggest ring with a hard press. Indexing down the block is a bit better but not perfect.
I’ve been trying to sort it all afternoon, taken one off tried the other then gone back to the other, same issue persists.
Any ideas?

Mucky cable outers.
Pull the cable out, blast through with spray lube via the little hose which fit perfectly in a cable outer and job done.

caught cable, sticky mech, limiter screws, height screw, bent drop out – so many possibilities.

As Sui said too. I’d assumed it was all set up right and as you’d swapped mechs, the cable could be the culprit.

Is the cable exposed in it’s run, if so use this to isolate the shifter. If you pull the cable outward does the problem still occur? Then it’s not the shifter.
Could be cable stretch, try re-indexing it. See park tools website if your unsure how to.

Old mechs do go sloppy too which doesn’t help, especially if the cable pull is hindered by muck.