  A version of the Schelling model on $\mathbb{Z}$ is defined, where two types
of agents are allocated on the sites. An agent prefers to be surrounded by
other agents of its own type, and may choose to move if this is not the case.
It then sends a request to an agent of opposite type chosen according to some
given moving distribution and, if the move is beneficial for both agents, they
swap location. We show that certain choices in the dynamics are crucial for the
properties of the model. In particular, the model exhibits different asymptotic
behavior depending on whether the moving distribution has bounded or unbounded
support. Furthermore, the behavior changes if the agents are lazy in the sense
that they only swap location if this strictly improves their situation.
Generalizations to a version that includes multiple types are discussed. The
work provides a rigorous analysis of so called Kawasaki dynamics on an infinite
structure with local interactions.
