  We present a one-parameter family $F_\lambda$ of transcendental entire
functions with zeros, whose Newton's method yields wandering domains,
coexisting with the basins of the roots of $F_\lambda$. Wandering domains for
Newton maps of zero-free functions have been built before by, e.g., Buff and
R\"uckert based on the lifting method. This procedure is suited to our Newton
maps as members of the class of projectable functions (or maps of the
cylinder), i.e. transcendental meromorphic functions $f(z)$ in the complex
plane that are semiconjugate, via the exponential, to some map $g(w)$, which
may have at most a countable number of essential singularities.
  In this paper we make a systematic study of the general relation (dynamical
and otherwise) between $f$ and $g$, and inspect the extension of the
logarithmic lifting method of periodic Fatou components to our context,
especially for those $g$ of finite-type. We apply these results to characterize
the entire functions with zeros whose Newton's method projects to some map $g$
which is defined at both $0$ and $\infty$. The family $F_\lambda$ is the
simplest in this class, and its parameter space shows open sets of
$\lambda$-values in which the Newton map exhibits wandering or Baker domains,
in both cases regions of initial conditions where Newton's root-finding method
fails.
