This will vary depending on your chosen hosted archive service.
For example, Transcault transfers data directly over the network to Microsoft Office 365 using the EWS protocol.
For extremely large data transfers, or where there is no existing API, a typical strategy is to store data on removable media in a neutral file format such as PST or EML files. In this case TransVault can be configured to package and name the exported data according to the conventions stipulated by the hosted vendor.
The removable media would then be shipped to the hosted service provider for ingestion.
To ensure peace of mind, Transvault creates a full report of exactly what data has been extracted from your system to be handed to the service provider.  This further acts as evidence of due diligence in support of future eDiscovery requests.