As best I can tell, the Kiln multiblock exposes all of its input, output, and fuel slots in a single inventory interface on each block/side. This effectively prevents being able to automate it with vanilla methods (such as hoppers), and most modded methods. Only piping/automation solutions that provide explicit, numbered inventory slot access would be able to do the trick, and that is very uncommon.

My suggestion is to make the sides behave as follows:

• Top: Expose the input slots
• Sides + Front + Back: Expose the fuel slot
• Bottom: Expose the output slots

This would allow using vanilla hoppers to automate the Kiln, as well as any modded solutions. Optionally, the Front side could expose the output slots (instead of the fuel slot) in addition to the Bottom doing so -- this wouldn't be of use to hoppers, of course, but would be slightly nicer/friendlier for modded automation methods.