The Dunwich rules don't seem to explicitly say this on page 6. Do these monsters stay when there is an investigator with them, or do they perhaps even opt to pursue one of lower sneak if they are adjacent but there is already one on its area? I figured they stay, but don't see a direct reference to this.
The basic monster movement rules state that monsters don't move when they are in the same space as an investigator.
There isn't any reason to think that the stalker addition in Dunwich changes this in any way.