These are Stirling cycle engines with a fluid piston acted upon by a heat-induced pressure oscillation resulting from cyclic heating and cooling of a fixed amount of a working gas (air) inside the engine.
While a fluidyne can be made to oscillate, and even to pump water, it does not perform well at low pressure under load because as the loading increases the engine uses more and more of its mechanical energy to modulate the pressure of the air in its hot head.
In simple terms, the difficulty appears to be that thermal expansion is offset by mechanical compression of the air resulting from the loading forces.
It may be possible to circumvent this problem behavior by operating the engine at significantly higher pressure and temperature, but when those conditions are acceptable, there are better options than fluidynes. To see some of my exploration of those options, go back to "Engines" and browse "Hydrodyne Engines" and "Hydradyne Engines".
I find air/water fluidynes unsuitable as a practical pump engine solution.