When you perform a transfer to another registrar, they will send you a confirmation message. If you approve the transfer on their side but take no further action, the transfer will complete by default after about seven days. However, after you approve the transfer with your new registrar, our registrar should* also send you a confirmation message that should allow you to approve or decline the transfer. If you approve the transfer there as well, it should complete much more quickly.
In most cases, transferring a domain near to or shortly after its expiration will not result in any problems or disruption in service; each TLD is managed through a central registry and their systems are smart enough to recognize that the domain has been renewed, even if it's through a different registrar. However, not all registrar's systems are capable of processing the transfer of expired domains, so even though it will probably be fine, to avoid the consequences of probably, we always recommend that you manage any transfers well in advance of domain expiration. If you encounter a problem with the transfer of an expired domain, we will probably be unable to help.
*Key word: should. Most of the time, they do. Sometimes they don't. We have no control over whether one is sent, no ability to resend it, and no access to information about whether one was sent, bounced, or got rejected as spam. That's not how it should work, of course, but nobody seems especially interested in how we think it should work.