It depends on the application, but the most likely scenario is you will have to configure all apps to use an ADFS trust instead of an Azure AD trust.
It's possible that some applications can simply continue to use Azure AD trusts and then Azure AD will handle federated authentication with ADFS, but this would complicate the login process a lot and make it more difficult to manage and troubleshoot. Also, adding ADFS means adding a potential point of failure, as in "if ADFS doesn't work you won't be able to log on to anything" (which is why ADFS is usually implemented with at least a two-servers farm).
Side note: you don't "migrate the domain to "ADFS Authentication" in Microsoft AD Connect"; you'll need to setup an actual ADFS farm (including a reverse proxy for publishing it externally) and then configure the domain for federated authentication in Azure AD.
I don't know the specifics of your scenario, but this seems quite a bit complex, especially if you don't really have good experience with ADFS (which, no offense intended, you don't seem to have); if the customer's reason for doing all of this is simply to use their own MFA solution, I'd strongly advise them to just enable Microsoft's MFA, or switch to one of the various cloud MFA solutions which can be integrated with Azure AD.