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We discuss precessing compact binaries on eccentric orbit with gravastar, black hole, neutron star, or boson star components. We derive secular evolution equations to second post-Newtonian--order accuracy, with leading-order spin-orbit, spin-spin, and mass quadrupole-monopole contributions. The emerging closed system of first-order differential equations evolves the pairs of polar and azimuthal angles of the spin and orbital angular momentum vectors together with the periastron angle. The secular dynamics is autonomous. We confirm numerically, that secular evolutions look like smoothed-out instantaneous evolutions over timescales where radiation reaction is negligible. The secular evolution of the spin polar angles and the difference of their azimuthal angles generates a closed subsystem. We study analytically this system for the particular cases of one spin dominating over the other and for black hole - boson star binaries of equal masses. In the first case known large flip-flops of the smaller spin are reproduced, when the larger spin is almost coplanar with the orbit. We also find new, quadrupole-induced flip-flops occurring when the neutron star with dominant spin has a quadrupolar parameter $w_1\approx 3$. Finally we analyze the evolutions of the spin angles numerically by comparing the cases when the black hole companion is either a gravastar, another black hole or a boson star with identical mass. We find that both the amplitude and period of the flip-flop are the largest, when the companion is a black hole. In the case of a boson star companion the frequency of the flip-flop increases significantly. Further, while in the case of gravastars and black holes a swinging-type azimuthal evolution occurs, with the spins of the components periodically overpassing each other, their sequence is conserved when the companion is a boson star.