Seamless mobility of mobile objects (MOs) such as containers or virtual machines cannot be guaranteed when MOs move across different networks of which the IP address domains are different. In this paper, we propose what is termed the live migration (LM) system and protocol that together provide seamless mobility for MOs connected to the Internet moving across different networks. The components of the LM system are the LM distributed hash table (DHT) system that stores the locator information of MOs and the LM Access Router, cooperating with the DHT system, which supports the mobility of MOs. The LM Access Router inserts the identifier of an MO into a packet and routes the packet to the LM Access Router to which the MO is associated. The LM system does not require modification of the network protocol stack of the host on which an MO is running, and it can support mobility with the IPv4 and IPv6 Internet protocols by introducing LM Access Routers into the network. We implemented the LM system on a testbed and conducted an experiment on the live migration of containers and virtual machines across networks. The findings showed that downtime of the live migration of a container to a host on a different network can enable the seamless live migration of MOs across networks.