  We construct exact, time-dependent, black hole solutions of
Einstein-Maxwell-dilaton theory with arbitrary dilaton coupling, $a$. For $a=1$
this theory arises as the four-dimensional low-energy effective description of
heterotic string theory. These solutions represent electrically charged,
spherically symmetric black holes emitting or absorbing charged null fluids and
generalize the Vaidya and Bonnor-Vaidya solutions of general relativity and of
Einstein-Maxwell theory, respectively. The $a=1$ case stands out as special, in
the sense that it is the only choice of the coupling that allows for a
time-dependent dilaton field in this class of solutions. As a by-product, when
$a=1$ we show that an electrically charged black hole in this theory can be
overcharged by bombarding it with a stream of electrically charged null fluid,
resulting in the formation of a naked singularity. This provides an example of
cosmic censorship violation in an exact dynamical solution to low-energy
effective string theory and in a case in which the total stress-energy tensor
satisfies all energy conditions. When $a\neq1$, our solutions necessarily have
a time-independent scalar field and consequently cannot be overcharged.
