The swinging only comes on for a few seconds and it would be out off the blue. I would be going down the road at various speed, 55,60,65,70 and it doesn't matter what speed I was at, it dies as quickly as it appears. The swing itself isn't too great maybe 6 inches either way. A trip down to Lymington, say 40 miles there and back, no more than half a dozon swayings.
The tyre pressure is 50 I think (might have been 55). The brakes are fine, both rim are cool as an ice cream after 40 miles.
Sorry new to towing anything so am really unsure of what to expect.