This paper explores the issues that arise in the context of the migration of a robotic pet between different embodiments and the associated design challenges. In the following, we describe the perceptions that a group of children have of a dinosaur character crossing the boundary between its robotic embodiment (the Pleo commercial pet), and its virtual counterpart on a mobile phone. We analyse the children's perceptions of, as well as emotional reactions to, the migration of this character, and show how seemingly subtle variations in the migration process can affect the children's perception on the character and its embodiments. Among other findings, gaps in the migration process, or perceived unresponsiveness, appeared to be accompanied by anxiety in the participating children. Based on our results, we point to yet unsolved design challenges for migration in interactions with embodied characters, and offer insights for migration implementation.