The turkeys are a little unique because you don't want to dry them out. I have a regular charcoal, water, dome smoker that I have owned some version of it for years. They don't make that style without the doors anymore.
If it gets cold outside, keeping heat in that thing can be a challenge later in the cooking cycle. I have a 13-14 lb bird (actually 2 of the same size) in mine for about 6 1/2 hours. I put it on overnight, so it is ready in the morning. I have been doing it for so long that I can just feel the top on it, and know when it is time to go to bed. It is outside on some patio bricks away from the house, and I can set up my brewing canopy if it looks like rain.
It might be tough to do one of those on a grill unless you can keep the heat around 225 and keep a shitload of water evaporating on it.