A weakness in your clientside hashing will make your site weaker to brute-force attacks, since it will reduce the number of hashes (or passwords) an attacker has to try (collisions in client-side hashes will too, but very negligibly for a good hash function). It's also impossible to recover from without relying on another form of authentication to re-establish trust. For many sites this means downgrading to single-factor.

Any hash upgrade mechanism can be abused by a (possibly MITM) attacker to change a user's password while leaving you and the user none the wiser that specifically this occurred. If you need to lock someone out while their phone is beeping at them over their bank account being emptied, while not even making it look like their password was changed, that sounds like a fun way.

Lastly it's virtually the same as plaintext, since any salt will be known by even just a passive attacker. A true MITM won't even have to brute-force the hash.

Conclusion: Might do harm, will do little good.