I need help getting LO to sleep in the crib.

LO is a few days shy of 3mos. He currently bedshares with us overnight. He naps in the swing or in a rock-n-play. He usually also spends his first bedtime sleep stretch in the rock-n-play, then comes into our bed at his first wake-up.

He's starting to get pretty smushed in the rock-n-play, and he's starting to get more mobile, so, we need to get him sleeping in the crib sooner rather than later.

Problem is that he often wakes up immediately if we try to put him down on the flat crib mattress, or he sleeps fitfully and wakes up much sooner than he usually does in the rnp.

Any tips for getting him to sleep in the crib? Or leads on a hammock-y bed similar to the rnp for bigger babies?