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We study extreme mass ratio inspirals (EMRIs) due to an infalling LightCompact Object (LCO) onto a generic class of stationary and axi-symmetricmassive compact objects (MCO - with or without a horizon). Using the quadrupolehybrid formalism we obtain a master formula for the evolution of the radius ofthe LCO and find qualitatively different behaviours depending on the geodesicstructure of the MCO. We then specialize the MCO to a black hole withsynchronised scalar hair (BHsSH). To allow a comparison with a highly spinningKerr BH, we consider BHsSH with dimensionless spin, $j = 1$. This yields twodistinct sequences of solutions. The first harbours Kerr-like solutions withmaximal hairiness of $\sim 10\%$. The corresponding EMRIs are Kerr-like, butthe cut-off frequency can be a few times smaller than in Kerr, yieldingwaveforms with quantitatively significant non-Kerrness. The second sequencelinks the extremal Kerr black hole to a mini-boson star with $j=1$. Here weobserve qualitative non-Kerrness, such as the non-monotonically increase of theangular velocity and stagnation endpoints, reflecting Kerr-unlike geodesicstructures.

EMRIs around $j=1$ black holes with synchronised hair