  We study two generalizations of classic clustering problems called dynamic
ordered $k$-median and dynamic $k$-supplier, where the points that need
clustering evolve over time, and we are allowed to move the cluster centers
between consecutive time steps. In these dynamic clustering problems, the
general goal is to minimize certain combinations of the service cost of points
and the movement cost of centers, or to minimize one subject to some
constraints on the other. We obtain a constant-factor approximation algorithm
for dynamic ordered $k$-median under mild assumptions on the input. We give a
3-approximation for dynamic $k$-supplier and a multi-criteria approximation for
its outlier version where some points can be discarded, when the number of time
steps is two. We complement the algorithms with almost matching hardness
results.
