If the sink drain starts running slow or becomes completely blocked up, the line would likely need to be power rodded. The plumber would remove the drainpipe p-trap or j-bend assembly and clean out those pipes. Then they would power rod the drain line going all the way to the larger drain. This should restore the full flow and the line should start draining properly.

Another way this could be cleaned out is with a kinetic air ram, or blow gun. The plumber would connect the plunger to the drain and close off other drains that would allow air to escape. The unit would be charged to build air pressure then the plumber could shoot different levels of pressure into the line to dislodge the blockage. If the pipes are too old, the plumber may advise against this procedure as it could potentially cause the pipes to break.