If I use one I put a little solvent or silicone oil on it, place in breech, then push in a little further with a wad cutter shaped pellet and fire both. I do not fire gun with a wool cleaning pellet only, I think it may allow the piston to fly forward too fast and damage the gun, much like firing without a pellet. This is for a springer, I would think with a SSP,PCP or pump up it would not matter.
If cleaning from the muzzle, take care to avoid raking the dowel across the ends of the rifling. Particularly if introducing an abrasive, even a supremely fine one like J-B Bore Brite. Any incidental grit that comes into contact with the dowel will also freely embed, making it a sort of sanding stick. A brass cup muzzle guide would be advisable.
A pull-through is a better approach in most cases, and one can be made from a piece of string trimmer line for free or nearly free. Melt a ball on one end and use a razor to cut a sharp point on the other end. A soda straw makes it easy to guide it through baffles or other muzzle device. Patches can be cut from a cotton T-shirt or flannel.