Given a place between two fields, the isotropy behaviour of Azumaya algebras with involution over the valuation ring corresponding to the place is studied. In particular, it is shown that isotropic right ideals specialise in an appropriate way. The treatment is characteristic free, and it provides a natural analogue to the existing specialisation theory for non-singular symmetric bilinear forms. The rest of the paper then deals with the case where the residue characteristic is different from 2. In that case we show that if the valuation ring is Henselian then isotropy can be lifted from the residue field to the fraction field of the valuation ring, and this can then be used to show that rationally isomorphic Azumaya algebras with involution over this Henselian valuation ring are already isomorphic. This then implies there is a notion of good reduction with respect to places for algebras with involution, just as for non-singular symmetric bilinear forms.