  Consider a distributed task where the communication network is fixed but the
local inputs given to the nodes of the distributed system may change over time.
In this work, we explore the following question: if some of the local inputs
change, can an existing solution be updated efficiently, in a dynamic and
distributed manner?
  To address this question, we define the batch dynamic CONGEST model in which
we are given a bandwidth-limited communication network and a dynamic edge
labelling defines the problem input. The task is to maintain a solution to a
graph problem on the labeled graph under batch changes. We investigate, when a
batch of $\alpha$ edge label changes arrive,
  -- how much time as a function of $\alpha$ we need to update an existing
solution, and
  -- how much information the nodes have to keep in local memory between
batches in order to update the solution quickly.
  Our work lays the foundations for the theory of input-dynamic distributed
network algorithms. We give a general picture of the complexity landscape in
this model, design both universal algorithms and algorithms for concrete
problems, and present a general framework for lower bounds. In particular, we
derive non-trivial upper bounds for two selected, contrasting problems:
maintaining a minimum spanning tree and detecting cliques.
