Even the mobile device is somewhere at home.
At home network mobile device has registered its HA, home address, which is fixed and linked to DNS record. This mobile device establishes also a Home Agent, which stores information about mobile devices whose permanent home address is in the home agent's network.
While away from its home network, a mobile device is associated with a temporary address, a CoA, care-of address, which is associated with the network the mobile node is visiting and establishes a Foreign Agent. If there is no foreign agent in the host network, the mobile device has to take care of getting an address and advertising that address by its own means.
Mobility in IPv6 specifies how a mobile device registers with its Home Agent and how the Home Agent routes datagrams to the mobile device through the tunnel between Home and Foreign Agents.