Is there any downtime while a domain is being transferred?
Transferring a domain between agents (registrars) only changes who manages the domain at the registry level — it doesn't, by itself, affect the domain's nameservers or DNS records.
If you keep the same nameservers throughout the transfer (that is, you don't simultaneously switch to another hosting or DNS provider), the site and email associated with the domain keep working normally, without interruption, no matter how long the registry-level transfer process takes.
Interruptions can only happen if, alongside the agent transfer, you also change the domain's nameservers to a different hosting provider — in which case the usual nameserver propagation times apply (see How long does nameserver propagation take?). We recommend keeping the two operations separate: finish the agent transfer first, then change nameservers afterward, if needed.