The trackway [towing-path] on the River Barrow changes from the east {left) bank to the west at Leighlinbridge and back again at Graiguecullen/Carlow.
It seems to me that there may have been some difficulties in getting horse-drawn boats from one side of the river to the other and I have found no evidence on how it was done, so here is some speculation instead.
Irish waterways history 
Greetings Brian.
Reading your post reminded me that I need to ring Jim Gill – who worked on the Barrow from 1945 until 1996. Doing so I asked if he knew the answer to this. It seems, at least if I understand him correctly, that he does:
At Leighlinbridge horsedrawn boats were taken up the east bank as far as the bridge, and then the boat was tied off and the horse released. It would then be walked over the bridge, while the towrope would be floated across and down the river to be collected some way downstream of the bridge on the west bank. It would then be tied off to the horse, and the boat would then be drifted/ferry-glided across the river.
The weir at Carlow made switching banks a rather more brutal affair. Jim tells me that here the boat would leave the lock on the west bank and be held against the weir’s catwalk. Then the horse would be released and walked over the bridge to wait on the east bank. The boat would be shifted by the combined brute of men on the bridge hauling on the towrope, and more men manhandling it from the catwalk. I guess the presence of the Grand Canal Company’s stores just above the bridge meant there were plenty of bodies about… Once on the east bank the horse would be reattached and the barge would then be conventionally dragged through the navigation arch and head on upriver.
While the above only covers the upstream passage of craft, I’d assume – and Jim didn’t suggest otherwise – that downstream bound craft reversed the above.