I assume the ACFT has already been trimmed to descend at a given rate.
BrATCO, not necessarily!
If a VNAV PATH descent is being used when the new direct is given the "level abeam XXX" and the FMC is updated for the direct VNAV PATH will (assuming now low on path for next constraint after) go to a cruise descent of 1,000 ft/min.
If the "level abeam XXX" constraint still applies then the ABM Waypoints have to be selected and the constraint reentered.
Hope that makes sense!
Of course, other modes such as level change or vertical speed can be utilised as appropriate.