1. A method for a computing system to facilitate dynamic migration of networked computing nodes, the method comprising: managing multiple communications sent to multiple computing nodes of a first computer network while the first computer network is in use, the multiple computing nodes being located at a first geographical location and each being associated with one of multiple distinct network addresses for the first computer network, the multiple communications each indicating at least one destination network address that is one of the multiple network addresses so as to specify at least one of the multiple computing nodes as being an intended destination for the communication, the managing of each of the communications including forwarding the communication over the first computer network to the at least one intended destination computing nodes for the communication based on the at least one destination network addresses indicated by the communication; after the managing of the multiple communications, and under control of one or more computing devices configured to dynamically and incrementally migrate 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, and for each of the multiple computing nodes of the first computer network: receiving 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 computer network, such that after the migration, the replacement computing node of the remote virtual 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 automatically associating 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 computer network; and after the migration of a subset of the multiple computing nodes, automatically managing one or more additional communications sent over the first computer network that each indicate a destination network address used by the 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, so that the configured gateway device forwards the additional communication to the remote virtual computer network for delivery to that replacement computing node.