This is a great process! Just be sure you’re not missing out on shy folks who take a while to warm up. They might be a great communicator but struggle at the whiteboard in front of strangers.
That is why the first step is introduction to the team members, and then next is pair programming with few of them on the team. This will "warm up" the shy candidate. I myself fall into this category, hence I came up with that sequence of steps :)