  This paper considers the slow motion of the shock layer exhibited by the
solution to the initial-boundary value problem for a scalar hyperbolic system
with relaxation. Such behavior, known as metastable dynamics, is related to the
presence of a first small eigenvalue for the linearized operator around an
equilibrium state; as a consequence, the time-dependent solution approaches its
steady state in an asymptotically exponentially long time interval. In this
contest, both rigorous and asymptotic approaches are used to analyze such slow
motion for the Jin-Xin system. To describe this dynamics we derive an ODE for
the the position of the internal transition layer, proving how it drifts
towards the equilibrium location with a speed rate that is exponentially slow.
These analytical results are also validated by numerical computations.
