  A Dynamic Graph Network is a graph in which each edge has an associated
travel time and a capacity (width) that limits the number of items that can
travel in parallel along that edge. Each vertex in this dynamic graph network
begins with the number of items that must be evacuated into designated sink
vertices. A $k$-sink evacuation protocol finds the location of $k$ sinks and
associated evacuation movement protocol that allows evacuating all the items to
a sink in minimum time. The associated evacuation movement must impose a
confluent flow, i.e, all items passing through a particular vertex exit that
vertex using the same edge. In this paper we address the $k$-sink evacuation
problem on a dynamic path network. We provide solutions that run in $O(n \log
n)$ time for $k=1$ and $O(k n \log^2 n)$ for $k >1$ and work for arbitrary edge
capacities.
