Les, I'm no expert but here's what I've figured out from studying old mines and what little I've found on the web and in print...
The methods vary depending on the type of shaft, the era, and the size of the mine.
A vertical shaft can use buckets to lift out the ore, but you'd only see this on very small, primitive mines.
Another method is to have a platform on the lift (also called a "cage") which is fitted with a short section of track. A single ore car is pushed by hand onto the lift from whatever level of the mine is being worked. The hoist lifts it to the surface, and then a worker pushes the ore car onto a track to be dumped, either into the ore bin or the waste pile. This is a simple method used on smaller and/or older mines, but it's not very efficient.
On a larger and/or more modern mine with a vertical shaft, the workers would dump the ore into a special ore car attached to the hoist cable, called a skip. The skip is pulled to the surface and dumps (usually) automatically.
A mine with an inclined shaft would generally use a skip designed for use at an angle rather than vertically. As near as I can tell these also dump automatically.
Exactly how the vertical and inclined skips dump isn't real clear to me, but it usually involves an extension of the rails that flips the skip over.
I've also seen some inclined skips with wider wheels on the back end, and narrower wheels on the front end. Apparently these allow the front end to drop down between the rails, which are wider at the dumping point.
That's what I decided to use for the inclined shaft of the Cliffside Mine:
The rails are wood fitted with styrene "angle iron".
You can see a good example of a vertical shaft headframe and skip here:
This is the "Death Valley Mine" near Cima, CA. It's a small mine of relatively recent vintage, with a homemade, steel headframe. It's remarkably intact, with the skip still in place. There was a large tube nearby made of steel drums welded together, which was apparently used as a chute.
The "Great Western Mine" in Gold Point, NV has an inclined shaft with a fairly large wooden headframe combined with a steel hopper:
You can see the steel headframe of the "Orleans Mine" there too, and it also has an inclined shaft.
There's a smaller wooden headframe and inclined shaft at the "Noonday Mine" in the Mojave desert:
The "Nivloc Mine" in NV has a vertical shaft and large steel headframe. If you look near the top of the headframe you can see the remains of the mechanism for dumping the skip. But there are also signs of some old 18" gauge tracks on the ground which seem to indicate that this mine originally brought up the ore cars on a lift.
Hope this helps!