Just as with Events and Event Hotels, you can give your sub-blocks their own cut-off date. At cut-off, the behaviour is the same as with the event-level cut-off date; the front-end for the sub-block will close, and bookings will no longer be possible with the appropriate accompanying message.
With sub-blocks, you have a couple of extra features. Notice above, there is also a drop-down labeled "Cut-off Action". With this drop-down, you can decide what will happen when the cut-off date arrives while there is leftover inventory in the sub-block. You can either leave the leftover inventory in the sub-block, or you can have it return to the main block for consumption there. Regardless, you do still have control over those rooms until your latest cut-off date is reached.
The other feature you can utilize is sending a sub-block's administrator an email to remind them of an incoming cut-off date for their sub-block, or that cut-off has arrived and booking is no longer possible. With these two emails, you can specify to whom they are addressed, and what the contents will actually say.