It's very rare that the client protocol changes dramatically and in an incompatible way, so typically nothing changes and no one notices.
I've seen a few rare cases where an update to the client library is required for improvements to the protocol, e.g. when MySQL switched to the new password/authentication format, but that was also backwards-compatible: the old format was used until an account's password was changed, password changes could be told to use the legacy format instead, and you could set the legacy format by default. The only issue that occurred (and it did occur) was old client libraries + new server password hashes, but, again, in this case it didn't affect existing installations/users/passwords.
I've seen a few rare cases where an update to the client library is required for improvements to the protocol, e.g. when MySQL switched to the new password/authentication format, but that was also backwards-compatible: the old format was used until an account's password was changed, password changes could be told to use the legacy format instead, and you could set the legacy format by default. The only issue that occurred (and it did occur) was old client libraries + new server password hashes, but, again, in this case it didn't affect existing installations/users/passwords.