It's a little hard to understand given the context. Typically, at larger firms, the process for switching roles varies by seniority. For simplicity, let's say there are "senior" roles and "junior" roles. Senior roles are usually filled via a skating or succession process (eg there isn't a job to bid on). Junior roles are generally filled, administratively at least, very similarly to an external hire. There's often an interview (albeit different formality) which is usually followed or preceded by due diligence by the new hiring manger.
I'm assuming the meetings you described were effectively proxies for interviews or fit. Assuming there is a fit, most organizations ensure that there is an amicable time frame established for the transition to allow the manager of the departing resource to maintain business continuity. This is generally a non-issue because employees in these roles are generally more fungible and .org. want to retain and promote talent. Of course, this assumes this isn't a critical time of year, there's some major activity going on where you are truly indispensable, or you've been in the role less than a year (or whatever Hr policy is)
I'd simply have a conversation with your manager about the situation to understand his/her perspective and take it from there.
