A domain registrar, a DNS provider and an email provider can all be three different companies. That distinction is what matters here. Changing the registrar only changes who manages the domain registration. It does not, by itself, tell Microsoft 365, Google Workspace, your web host or another mail provider to stop handling your email.
Managing more than 300 domains has made me quite strict about this. Before I move a domain, I want to know exactly where its nameservers point and where its DNS zone is hosted. Email is one of those services that can look completely unrelated to a registrar transfer right up until an old DNS zone disappears.
The important question is not “who is the registrar?”Ask “who is hosting the DNS zone that contains my mail records?”. If that DNS service survives the transfer unchanged, the email service usually does too.