The procedure will vary from Airport to Airport and different ATS procedures.
If there is SMR or ASMGCS in use the Tower controller will continue to monitor your movement on the ASMGCS or SMR to make sure you clear the runway and ILS Sensitive areas , but normally the handoff to Ground is to make sure that you keep taxiing forward so that GMC may give you the next taxi instruction. If there are standard taxi routes published this is not as critical as where GMC may require you to turn off the RET to parallel the runway instead of moving in the direction you may think best to your stand.
Generally if Tower has transferred you to GMC once vacated they do this so that GMC can keep you moving on the taxiways without you slowing too much and Tower will use the ASMGCS or SMR to monitor when you are clear of the Sensitive areas. Same as Approach may hand you to ACC early to make sure you get continuous climb.
Was it clear which taxi route you were required to take when vacating the runway after the briefed RET or did you require a turn onto another taxiway shortly after you vacated? If the latter was the case then the local Tower procedures would state that transfer of communication is to occur once runway vacated so that the GMC can give the next taxi instructions and Tower monitors the progress on the SMR. If GMC can't give taxi instructions then this is coordinated between GMC and TWR reference the sensitive areas