1. A computer-implemented method, comprising: forwarding a communication over a first computer network to an intended destination computing node based on a destination network address indicated by the communication, the intended destination computing node being one of multiple computing nodes of the first computer network that are at a first geographical location and that are each associated with one of multiple network addresses of the first computer network; after the forwarding of the communication, dynamically migrating the first computer network to a remote virtual second computer network at a second geographical location while the first computer network continues to be in use, the dynamic migrating of the first computer network including, for each of the multiple computing nodes of the first computer network: receiving, by one or more configured computing systems, an indication of migration of the computing node of the first computer network to a distinct replacement computing node at the second geographical location that is part of the remote virtual second computer network, such that after the migration, the distinct replacement computing node of the remote virtual second computer network replaces the migrated computing node as part of the first computer network and uses the network address of the migrated computing node for ongoing operations of the first computer network; and associating, by the one or more configured computing systems, the network address of the migrated computing node on the first computer network with a gateway device on the first computer network that is configured to represent computing nodes of the remote virtual second computer network during the dynamic migrating of the first computer network; after the migration of a subset of the multiple computing nodes, managing, by the one or more configured computing systems, one or more additional communications sent over the first computer network that each indicate a destination network address used by the distinct replacement computing node for one of the computing nodes of the subset, the managing of each of the additional communications including forwarding the additional communication over the first computer network to the configured gateway device, the additional communication indicating a destination network address used by the replacement computing node for one of the computing nodes of the subset; and forwarding, by the configured gateway device, the additional communication to the remote virtual second computer network for delivery to the replacement computing node for the one computing node of the subset.