  We study the early time dynamics of bimodal spin systems on $2d$ lattices
evolving with different microscopic stochastic updates. We treat the
ferromagnetic Ising model with locally conserved order parameter (Kawasaki
dynamics), the same model with globally conserved order parameter (nonlocal
spin exchanges), and the voter model. As already observed for non-conserved
order parameter dynamics (Glauber dynamics), in all the cases in which the
stochastic dynamics satisfy detailed balance, the critical percolation state
persists over a long period of time before usual coarsening of domains takes
over and eventually takes the system to equilibrium. By studying the
geometrical and statistical properties of time-evolving spin clusters we are
able to identify a characteristic length $\ell_p(t)$, different from the usual
length $\ell_d(t) \sim t^{1/z_{d}}$ that describes the late time coarsening,
that is involved in all scaling relations in the approach to the critical
percolation regime. We find that this characteristic length depends on the
particular microscopic dynamics and the lattice geometry. In the case of the
voter model, we observe that the system briefly passes through a critical
percolation state, to later approach a dynamical regime in which the scaling
behaviour of the geometrical properties of the ordered domains can be ascribed
to a different criticality.
