Clinkers form when you get your firebox temps up to around 2500Degrees F. What happens is the heat liquifies the metals and minerals in the wood and forms clumps of the metal. These metals are mixed with ash to make clinkers. Seems to be common when burning elm, but I have never burnt elm and I have found them in my stove, so I just guess the mineral content your wood was grown in has a big part to play in how much clinkers you end up with.
I mostly burn oak and pine, and I commonly get them from oak. Probably because oak makes a big coal bed and pine doesn't. If I let it go a bunch of reloads without stirring up the ash in front, I get larger ones.
I've been getting it with ash and red maple. I think it is Ash melted together. I can see ash melt in the stove sometimes if there's a piece dangling in the middle of a flame. The lower air inlets get the coals plenty hot to melt ash.
I only get them during a cold spell when I’m feeding new wood faster than the coals are burning away - usually because I haven’t emptied the ash pan and not as much air is coming up through the grates. As said before, the ash gets hot enough to fuse into the clinkers.
I see them as the furnace’s way of scolding me that I need to empty my ash pan more often.
They're problematic in coal burners because they can block the grates above the bottom air intakes. Wood burners don't see them as more than a curiosity because ours are softer and we don't have air intakes there.
No, I still have ash block my vents. Clinkers and bark ash like Ash bark ash keep air flow from going straight back more than a couple inches. That's why I prefer barkless Oak or Osage which leave a fine powdery ash that doesn't clump or fuse easily.