  Recently it has been shown that when an equation that allows so-called pulled
fronts in the mean-field limit is modelled with a stochastic model with a
finite number $N$ of particles per correlation volume, the convergence to the
speed $v^*$ for $N \to \infty$ is extremely slow -- going only as $\ln^{-2}N$.
In this paper, we study the front propagation in a simple stochastic lattice
model. A detailed analysis of the microscopic picture of the front dynamics
shows that for the description of the far tip of the front, one has to abandon
the idea of a uniformly translating front solution. The lattice and finite
particle effects lead to a ``stop-and-go'' type dynamics at the far tip of the
front, while the average front behind it ``crosses over'' to a uniformly
translating solution. In this formulation, the effect of stochasticity on the
asymptotic front speed is coded in the probability distribution of the times
required for the advancement of the ``foremost bin''. We derive expressions of
these probability distributions by matching the solution of the far tip with
the uniformly translating solution behind. This matching includes various
correlation effects in a mean-field type approximation. Our results for the
probability distributions compare well to the results of stochastic numerical
simulations. This approach also allows us to deal with much smaller values of
$N$ than it is required to have the $\ln^{-2}N$ asymptotics to be valid.
