Question about moving Domain to new host

Hey guys my client has a site that is hosted through go-daddy - they are being charged an outrageous monthly amount for a total crap plan. I convinced them to set up an account with a better web host. I was wondering if anyone knows what happens to all the emails that she has received in her go-daddy account.

If I change the name servers will all the emails are in her old hosting account disappear?

Should I back them up somehow before changing the Name Servers?

 

All the email will be gone if you do not do backup to transfer it to new host email.

 

Whatever you consider very important must have back up taken befoe changing host.