  Extreme mass-ratio inspirals detectable by the future Laser Interferometer
Space Antenna provide a unique way to test general relativity and fundamental
physics. Motivated by this possibility, here we study in detail the EMRI
dynamics in the presence of a spinning secondary, collecting and extending
various results that appeared in previous work and also providing useful
intermediate steps and new relations for the first time. We present the results
of a frequency-domain code that computes gravitational-wave fluxes and the
adiabatic orbital evolution for the case of circular, equatorial orbits with
(anti)aligned spins. The spin of the secondary starts affecting the
gravitational-wave phase at the same post-adiabatic order as the leading-order
self-force terms and introduces a detectable dephasing, which can be used to
measure it at $5-25\%$ level, depending on individual spins. In a companion
paper we discuss the implication of this effect for tests of the Kerr bound.
