Change is constant, especially with your team. People leave, people come. Usually it’s not a big deal, but sometimes you have to transition a key person. Success then comes down to two things: training and expectations.
Training is more administrative than leadership, so it doesn’t need a lot of attention here. Suffice it to say the more training you can do with the outgoing person, the better.
Expectations, though, take some thought. Are they the same for the new person? Sometimes you end up with less experience and you want to lower expectations. Sometimes a new hire is a chance to up your game. No matter what you do, though, your team, and others, will expect the same things from the new person.
If you want transitions to succeed, and new people to acclimate quickly, you have to be very deliberate about thinking through what your expectations are, and then making sure your team has the same expectations. That way, the new guy or gal doesn’t get conflicting signals.