This paper investigates user perceptions of continuous identity as agents migrate between different embodiments. It reports an experiment seeking to establish whether migrating or not migrating the interaction memory of the agent would affect the user's perception of consistent agent identity over different embodiments. The experiment involved a treasure hunt in which a virtual agent migrated from a screen to a mobile phone in order to accompany a user while they searched for clues. A total of 45 subjects took part in three different conditions with 15 subjects in each. The outcome showed that the presence of memory affected the competence users ascribed to the virtual agent. However it had no significant effect on a strong perception of consistent identity across multiple embodiments.