The Brayton Cycle (best known as the working principle of the jet engine) potentially lends itself to solid fuel because it allows continuous combustion.
If you look at a jet engine you'll see (1) a big air compressor at the front, (2) a continuous fire in the combustion chamber, which adds energy to the compressed air, but crucially, no further increase in pressure(*), only an increase in volume (aka isobaric expansion), and (3) a way of recovering enough energy in a turbine to drive the compressor.
(*) If the combustion stage does add pressure, the compressor stops : this is called a "compressor stall", so the art of running a Brayton engine is to stay in the sweet spot between "flameout" and "compressor stall" which was a real problem for WW2 jet pilots, requiring careful manual throttle control especially on takeoff.
Any energy left over can be used as you want - thrust in the rapidly moving hot exhaust in an aircraft (or bizarrely, ship!), or use a bigger turbine at (3) and use its excess shaft power to drive a propeller, mill, pump, generator or whatever.
(photo of the Lucy Ashton, built 1887, converted from steam to jet power 1951)
Hypothetically, because of the continuous combustion stage, you could apply solid fuel such as wood, coal or even peat. The latter was developed in Caithness, Scotland in the 1950s, alongside sodium-cooled fast breeder reactors.
If you don't have the machining ability to create sufficiently efficient fan or centrifugal compressor and turbine, these can be implemented as piston and cylinder, as per George Brayton's original Ready Motor though this used gas or oil as fuel.
The fast reactors were more successful... I can find very little information on this scheme or the reasons for its failure : possibly the abrasive nature of ash content in the exhaust caused problems for the turbine; it would certainly be problematic for piston wear.