  In this paper we investigate the consequences of phantom crossing considering
the perturbative dynamics in models with interaction in their dark sector. By
mean of a general study of gauge-invariant variables in comoving gauge, we
relate the sources of instabilities in the structure formation process with the
phantom crossing. In order to illustrate these relations and its consequences
in more detail, we consider a specific case of an holographic dark energy
interacting with dark matter. We find that in spite of the model is in
excellent agreement with observational data at background level, however it is
plagued of instabilities in its perturbative dynamics. We reconstruct the model
in order to avoid these undesirable instabilities, and we show that this
implies a modification of the concordance model at background. Also we find
drastic changes on the parameters space in our model when instabilities are
avoided.
