I run a lot of MDF base and the corners are always where the trouble shows up, so I've been down this road plenty. What you're seeing is ussually not the trim moving. It's the wall. Drywall inside corners are rarely actualy square, and even if the base sits tight against the drywall, there's a hollow spot or a bow behind it that opens up once you look at it from an angle. Door casings look fine because you're caulking against jamb material that's dead flat and rigid, not a wall that waves in and out by a sixteenth here and there. MDF does move a little with humidity, but on baseboard it's mostly the length that changes, not the profile, so it's rarely what's opening a mitre or a cope. If the gaps are along the top edge where the board meets the wall, that's almost always the wall not being flat. I scribe or shim for the worst spots before anything gets nailed off permanently. For the caulk itself, the trick is a small bead and a wet finger. Cut the tip tiny, run a thin bead into the gap, then drag it smooth with a finger dipped in water so the excess wipes off onto your rag instead of smearing across the paint. If you try to fill a gap that's too wide with caulk alone it'll shrink back and look sketchy no matter what you do. Anything over about a sixteenth wide I'd rather fill first and caulk over. One more thing on pre-coped stock: those copes are usually cut to a theoretical perfect fit, and no inside corner in a real house is perfectly square. Even a good cope needs a touch of filing or a shaving taken off the back angle to sit tight. I check every cope against its actual corner before it goes up, because the factory one rarely drops right in without a little help.
If the gaps are small and consistent, a thin tooled bead will look crisp once it dries. If they're wandering in and out, it's the wall, and no amount of caulk will hide it from a raking light.