From mosquitto.conf, the following options (among many others) exist for bridging.
This variable marks the start of a new bridge connection. It is also used to give the bridge a name which is used as the client id on the remote broker.
Set the client id for this bridge connection. If not defined, this defaults to 'name.hostname', where name is the connection name and hostname is the hostname of this computer.
If both the configuration options are specified and are different, then which ID applies to the remote broker?