  The zero modes and current carrying capability of a cosmic string formed at
one phase transition can be modified at subsequent phase transitions. A new,
generalised index theorem is derived that is applicable to theories with
multiple phase transitions. This enables us to investigate the fate of string
zero modes during sequences of phase transitions in a variety of models.
Depending on the couplings that the breaking introduces, the zero modes may be
destroyed and the superconductivity of the string removed, and thus vortons
dissipate. We discuss the features of the theory that are required to produce
this behaviour and consider the implications of spectral flow.
