This is not something that is easy to do in SolidWorks. There have been any number of posts. I think that at least one person has had good luck, but I don't remember what trick they used. Try looking at threads in the "More Like This" box to the right or searching on "moving a chain" or similar phrases. You'll find a whole lot of posts where people fight with the problem. There must be a pony in there somewhere!
The problem of mates flipping is also very common. The usual solution is to move your parts more slowly.