The coevolutionary dynamics in finite populations currently is investigated in a wide range of disciplines, as chemical catalysis, biological evolution, social and economic systems. The dynamics of those systems can be formulated within the unifying framework of evolutionary game theory. However it is not a priori clear which mathematical description is appropriate when populations are not infinitely large. Whereas the replicator equation approach describes the infinite population size limit by deterministic differential equations, in finite populations the dynamics is inherently stochastic which can lead to new effects. Recently, an explicit mean-field description in the form of a Fokker-Planck equation was derived for frequency-dependent selection in finite populations based on microscopic processes. In asymmetric conflicts between two populations with a cyclic dominance, a finite-size dependent drift reversal was demonstrated, depending on the underlying microscopic process of the evolutionary update. Cyclic dynamics appears widely in biological coevolution, be it within a homogeneous population, or be it between disjunct populations as female and male. Here explicit analytic address is given and the average drift is calculated for the frequency-dependent Moran process and for different pairwise comparison processes. It is explicitely shown that the drift reversal cannot occur if the process relies on payoff differences between pairs of individuals. Further, also a linear comparison with the average payoff does not lead to a drift towards the internal fixed point. Hence the nonlinear comparison function of the frequency-dependent Moran process, together with its usage of nonlocal information via the average payoff, is the essential part of the mechanism.