ABSTRACT Fuzzy dark matter of ultralight axions has gained attention, largely in light of the galactic scale problems associated with cold dark matter. But the large de Broglie wavelength, believed to possibly alleviate these problems, also leads to fluctuations that place constraints on ultralight axions. We adapt and extend a method, previously devised to describe the effect of gaseous fluctuations on cold dark matter cusps, in order to determine the imprints of ultralight axion haloes on the motion of classical test particles. We first evaluate the effect of fluctuations in a statistically homogeneous medium of classical particles, then in a similar system of ultralight axions. In the first case, one recovers the classical two body relaxation time (and diffusion coefficients) from white noise density fluctuations. In the second situation, the fluctuations are not born of discreteness noise but from the finite de Broglie wavelength; correlation therefore exists over this scale, while white noise is retained on larger scales, elucidating the correspondence with classical relaxation. The resulting density power spectra and correlation functions are compared with those inferred from numerical simulations, and the relaxation time arising from the associated potential fluctuations is evaluated. We then apply our results to estimate the heating of discs embedded in axion dark haloes. We find that this implies an axion mass $m \gtrsim 2 \times 10^{-22} \, {\rm eV}$. We finally apply our model to the case of the central cluster of Eridanus II, confirming that far stronger constraints on m may in principle be obtained, and discussing the limitations associated with the assumptions leading to these.