If I move a client’s website to Webflow and their emailing hosting is provided by Microsoft or any other third-party email provider. Do I need to configure anything else or will the email continue to work after I have changed over to Webflow hosting?
This depends on the current solution…
If it’s hosted on something which has DNS management (such as cPanel/Plesk) and the DNS settings are managed there then you’ll almost certainly be (and should be) cutting this service and therefore disrupting the email service.
If DNS is managed at the domain provider or delegated to a dedicated DNS service then you should just be changing A/CNAME records and therefore the MX records (controlling email servicing) are unaffected.
In the first scenario you mentioned, if I cut the old web hosting service off once migration is complete and the DNS is managed within CPanel therefore disrupting email. Would I need to input the MX records at the domain registrar DNS and it would work again?
There may also be some verification data in the records like TXT records so these may also be required. Depends on the mail service (or any other service that may be in use somewhere).