Will past versions of that app signed with the old account continue to
work?
Yes. When you run an app, Gatekeeper checks that it was valid at the time that it was signed. It’s possible for Apple to block a validly signed app from running, but that uses a different path. That’s why, for example, revoking Developer ID certificates is a manual process.
How do I transfer the app ID?
Are you sure it has an App ID? A Developer ID app only needs an App ID if it uses a restricted entitlements, that is, an entitlement that must be authorised by a provisioning profile.
Regardless of the App ID story, when the new developer’s app ships it will have a different DR, which can cause problems for things like the file-based keychain and the built-in firewall. See TN3127 Inside Code Signing: Requirements.
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Quinn “The Eskimo!” @ Developer Technical Support @ Apple
let myEmail = "eskimo" + "1" + "@" + "apple.com"