I transferred couple of domains from GoDaddy to Moniker on the exact expiration day and it went fine. I transferred many domains from GoDaddy in their last 10 days before expiration and it went fine all the time.
PS: My advice is based on my recent experience with transferring from GoDaddy.
5 days is too long already, if you and the recipient are both online then do the transfer wait for recipient to be online and push happens almost real time within minutes only, so don't worry about it.
Zona said:
I have a name at GoDaddy which will expire on November 22 (5 days from now).
If I initiate a transfer to another registrar today and the name expires prior to going through the transfer process what happens to the name?