Email downtime during propogation

I have set up new websites for clients but never transferred existing ones to a new host. I have been asked by a client to move over to a new web host and I would like to know what I can expect during DNS propogation. Will visitors still be able to view the website if I am still signed up with the previous web host? What will happen to the emails that are connected to the domain name? Is there any way to stop them from bouncing?
Any help and advice will be really appreciated…