Given that it is now possible to observe the interference of molecules with comparatively numerous internal degrees of freedom, we might wonder what role these extra degrees of freedom play in the traditional double-slit experiment. Here we seek to understand the effect of these degrees of freedom in an Aharonov-Bohm type experiment. We find, perhaps unsurprisingly that an ionised object can exhibit the Aharonov-Bohm effect in a manner which is independent of its internal structure. Meanwhile, the Aharonov-Bohm type effects involving objects with magnetic, or electric dipole moments (the Aharonov-Casher and He-McKellar-Wilkens effects, respectively) depend explicitly upon the structure. When—due to some specified dynamics of the internal variables—an object's dipole moment changes during the experiment, we find that the corresponding phase shift depends upon the time average and variance of the internal dynamics. Furthermore, given that the internal dynamics are, to an extent, dependent upon the centre of mass motion there is also a small, transient loss of coherence associated with each internal degree of freedom.