  We study extreme mass ratio inspirals (EMRIs) due to an infalling Light
Compact Object (LCO) onto a generic class of stationary and axi-symmetric
massive compact objects (MCO - with or without a horizon). Using the quadrupole
hybrid formalism we obtain a master formula for the evolution of the radius of
the LCO and find qualitatively different behaviours depending on the geodesic
structure of the MCO. We then specialize the MCO to a black hole with
synchronised scalar hair (BHsSH). To allow a comparison with a highly spinning
Kerr BH, we consider BHsSH with dimensionless spin, $j = 1$. This yields two
distinct sequences of solutions. The first harbours Kerr-like solutions with
maximal hairiness of $\sim 10\%$. The corresponding EMRIs are Kerr-like, but
the cut-off frequency can be a few times smaller than in Kerr, yielding
waveforms with quantitatively significant non-Kerrness. The second sequence
links the extremal Kerr black hole to a mini-boson star with $j=1$. Here we
observe qualitative non-Kerrness, such as the non-monotonically increase of the
angular velocity and stagnation endpoints, reflecting Kerr-unlike geodesic
structures.
