the customer can immediately slurp data out of Provider A via a dedicated connection, shuttle it through the owned servers, then spurt it up into Provider B.
Most cloud providers go out of their way to dissuade you from doing this. For instance, Amazon won't charge you to load data in to their cloud storage, but there is a fee when you pull it out.
I also take umbrage at 'immediately'. You can immediately start the process, but it can take days or weeks to transfer a large dataset from one DC to another, even if they are yards apart and have great fat wads of fibre connecting them.