The influence of the initial fluctuations on the onset of scaling in t
he quench to zero temperature of a two dimensional system with conserv
ed order parameter is analyzed in detail with and without topological
defects. We find that the initial fluctuations greatly affect the way
scaling is reached, while the number of components of the order parame
ter does not play a significant role. Under strong initial fluctuation
s the preasymptotic linear behavior is replaced by the mean field beha
vior of the large-N model, producing the multiscaling to standard-scal
ing crossover also in the physically interesting cases of systems with
a small number of components.