Depending on how long you intended smoking it you are likely to have a food safety issue by cold smoking the meat for any length of time after it is cooked. When smoking the meat while cooking the meat is effectively in a sterile (hot) environment and the heat continues to increasingly prevent bacterial growth. If you have previously cooked the meat the sheer act of pulling it (unless you do this in sterile conditions) is going to introduce bacteria throughout the meat mass from your hands, the air, implements, plates etc which are not going to be subsequently killed by the application of cold smoke. Some will argue that the smoke itself is antibacterial, which may be true to a point, however even if it is you would not be able to guarantee that sufficient concentrations of it will come in contact with all of the surfaces of the meat.
It would be less of an issue if you were to keep the cooked meat cold below 4 C (-40 F) while it was being smoked however smoking at anything approaching room temperature for more than a short period would not be advisable.
Following on from a previous thread. Maybe this would be the perfect situation for using Liquid Smoke.