...so will the CC SDR# move on its own under the control of TTT if the control channel changes?
Yes. TTT uses the 'Main carrier' element in the SYSINFO PDU to determine what the main carrier (and where MCCH) is for a network (MNC)
If that value changes, then TTT will switch to that newly defined (main) carrier.
So long as the new (main) carrier that has been switched to 'Main carrier' element in the SYSINFO PDU reflects this new change then it should stay, otherwise it will switch to whatever it says the main carrier is.
'D-Release' PDUs also point to the control channel (MCCH) carrier when a call is ended so MS's end up back at the MCCH on the right carrier.
So if MCCH changes carrier while a MS is on a call (on a different carrier), it will switch back to correct carrier for MCCH.
Currently I don't use 'D-Release' for that purpose.
I have never seen MCCH change carrier myself, only heard from one other person about it and they never got back to me about it. Maybe TTT does the job. It should work OK in theory.
For further clarification, I'm not sure what the control channel is on my local network of around 6 different freqs...
As stated above, TTT will determine the main carrier of any carrier selected and switch (CC SDR#) to it.
TETRA Demodulator also shows the main carrier of the selected TETRA carrier.
In SDR# side panel, It is shown (as 'Main: zzzz xxx.xxxx MHz') with a blue link. The current selected carrier is shown as 'Current: zzzz xxx.xxxx MHz'. 'zzzz' = carrier #.