We've had this issue with a 2010 and 2013 Equinox, towing behind four different coaches. I think I've finally got it figured out! I have a drop down hitch to my Blue Ox tow bar; both the drop down and the Blue Ox has some play at the receiver. I think that play gets the car started going back and forth and then it gets pretty violent depending on your speed. I had a welder put a couple of beads on each piece that goes into the receiver, I used a grinder to take some off until it fits tight. About the same time I had my brainstorm, I had the front end aligned on the Equinox to rule that out (no problems there). The mechanic told me he's seen a lot of problems with the electronic traction control on the Equinox while pulling four wheels down. So now we turn off the traction control just before pulling the dash fuse. So far, with limited testing (a few hours driving, but purposely trying to duplicate the problem) this has worked! Next month will be the big test with two months on the road.
Instead of welding and grinding you can buy this: https://www.amazon.com/dp/B015TLZXB0/ref=cm_sw_r_taa_eTqyyb9HBMFCF.