My miner/LJ characters always use a pack llama and a pack horse. The llama is short enough that oftentimes I can just drag and drop stuff onto its legs that are showing under the horse's belly. Other times I just do as someone else suggested and use Control Shift. The bars for both animals are usually sufficiently offset so I can see at least a little of the one that is underneath to be able to drag and drop stuff onto it. If not, I just right click the top one to get to the one underneath.
If you're just walking along while mining or lumberjacking and not recalling between each spot, you can also open the backpacks of each pack animal and leave them open on your screen. Then you can just drag stuff to them.
I haven't experimented much with doing the same stuff in KR. It appears you can't use Control Shift to bring up the animals' bars and then drag and drop stuff onto one of them. So I think that in KR you have to drag and drop onto whatever you can see of a particular animal or click on its bar to open its backpack and/or walk along with the animals' backpacks open on your screen. I notice that it's much easier in KR to see parts of both a pack horse and a pack llama--their heads are offset sufficiently that I think you can probably just drag stuff onto one head and let go and have it end up in the right backpack.
If none of this works, then your best bet is to have separate stay and follow commands for the pets so it's easier to order them around.
