  We study the dynamics of the synchronization transition (ST) of
one-dimensional coupled map lattices. For the Bernoulli map it was recently
found by Ahlers and Pikovsky (Phys. Rev. Lett. {\bf 88}, 254101 (2002)) that
the ST belongs to the directed percolation (DP) universality class. Spreading
dynamics confirms such an identification, only for a certain class of
synchronized configurations. For homogeneous configurations spreading exponents
$\eta$ and $\delta$ are different than DP exponents but their sum equals to the
corresponding sum of DP exponents. Such a relation is typical to some models
with infinitely many absorbing states. Moreover, we calculate the spreading
exponents for the tent map for which the ST belongs to the bounded
Kardar-Parisi-Zheng (BKPZ) universality class. Our estimation of spreading
exponents are consistent with the hyperscaling relation. Finally, we examine
the asymmetric tent map. For small asymmetry the ST remains of the BKPZ type.
However, for large asymmetry a different critical behaviour appears with
exponents being relatively close to the ones of DP.
