  We consider a class of discrete-time one-dimensional quantum walks,
associated with CMV unitary matrices, in the presence of a local field. This
class is parametrized by a transmission parameter $t\in[0,1]$. We show that for
a certain range for $t$, the corresponding asymptotic velocity can be made
arbitrarily small by introducing a periodic local field with a sufficiently
large period. In particular, we prove an upper bound for the velocity of the
$n$-periodic quantum walk that is decaying exponentially in the period length
$n$. Hence, localization-like effects are observed even after a long number of
quantum walk steps when $n$ is large.
