(Added in edit): I don't know why, but I didn't spot first time around the last line:
Is it that limits and colimits are more like their counterparts for manifolds?
which is odd, because that's the subject of a little theorem I've proved which can be found on the nLab:
Essentially, if you want to preserve those limits and colimits that already exist in the category of manifolds, then you need to work in the category of Hausdorff Frölicher spaces. When you enlarge that category (say to Frölicher spaces or to Diffeological spaces, or to sheaves) then you add in stuff "in the gaps" and create new limits or colimits that disagree with the ones that you had before (in these cases, it's almost always colimits, but if you take the "maps out" view, it will be limits). So that question isn't really a sensible one to ask of Diffeological spaces as you've already lost some colimits. I suppose you can try to do a bit of damage limitation ...