While not the only cause, it is well known that tension variations cause a web to move in the cross-machine direction. The web displacement or movement can be in either direction, to the operator size or to the back side. With many webs, the displacement can cause severe problems.
Some of the problems I have seen are listed. On coaters, the coating may build up at the edges of the web. If the web moves, its edge will cross over the buildup and break.
If the web is slit, the slitting may cut off any evidence of web wandering. However, if the web is printed, laminated or embossed prior to slitting, the displacement may damage the product. I once worked with a construction product with bubbles. Web displacement caused the slit to cut the bubbles, rather than on the flat track between the bubbles.
Not all web-displacement problems are caused by tension variations. An easy method to determine whether tension variations are the source of the problem is to determine if the displacement happens only when speed is changed. An additional test is to see if a change in tension setpoint causes the displacement while running at a steady speed.
If tension variations cause web displacement, check for mechanical problems such as roller alignment. The drive will probably require improvements in tension regulator tuning.