I have a mill head like this stripped bare on my bench right now. Here are some points of wear to inspect...
First, the 2 variable pulley halves that slide up/down (the other halves are fixed). There's one on the motor shaft and another on the spindle pulley hub. The common (very) wear points for these pulley halves are the shaft keys which are made of brass or even plastic and get chewed up by the steel shafts. The other are the oil-less bushings inside the pulley bore they slide up/down on which can be a PTFE impregnated metal shell or even a plastic bushing. These simply wear out and your pulleys get all sloppy and wobbly.
Second, the spindle pulley hub and the spindle gear hub. These mesh together when you shift the mill into high speed range. As you shift into high speed range what's happening is the shift lever is raising the spindle gear hub up, into the spindle pulley hub. Each hub has wedge shaped teeth that mesh together so the closer they come together the tighter they mesh until there's no slop at all. These teeth wear and on most heads there's an adjustment to tighten them as they wear. Eventually they wear to the point that the hub teeth bottom out in each other then it's game over and time for new hubs.
It could be other things like bearings, but the above two wear issues are far more common so I'd start by inspecting those first. If they require repair (they are wear parts that need replacing at some point, like belts) I wouldn't let it slide or other parts of the head will begin to self destruct.