In light of modern browsers phasing out security certificates signed using the SHA1 hash algorithm, we're busy replacing all our SHA1 certificates with SHA2. Generally we could simply replace certs for these primarily internal-use web apps in the evening or on the weekend, when there was little to no traffic.
What would happen if I were, unknowingly, in the middle of an encrypted session and the certificate for the domain was replaced?
To be on the safe-side, we advised our clients that they could assume users in mid-session during this change could see an interruption of their session, and possible loss of any data not yet stored in the database. If I were in mid-session during a cert replacement, could I assume that when I loaded the next page, after replacement of the cert, my browser would see a different signed cert than my session was established with, and cause the session to "freak out". I would expect all browsers would deal with this situation in a similar way, but please enlighten me if I am mistaken.
I have spent a fair amount of time searching for more specifics on how browsers would deal with this scenario, but I haven't had much luck finding general or technical information. I am really curious, and have decided to post this question in hopes of getting an answer which answers the Q concisely, with reference to some credible sources to validate.