I've not seen any documentation for this particular circumstance, however, sailing 'up' the channel (as the Spanish Armada had discovered) was risky, not least because of the bad weather that was present as the dispatches arrived on the English coast.
The vessels (HMS Pickle and HMS Entreprenante) that were given the dispatches were small and relatively fast but were not powerfully armed. The reason that multiple dispatches were sent on different ships was that these vessels could be lost (because of the natural risks at sea) or intercepted. While the British Navy had the main French fleets bottled up in their ports, there were still numerous French cruisers (both regular navy and privateers) waiting to dart out into the channel to pick off any British ships that they could. A small, lightly armed dispatch boat would have been a tempting target.
By landing the communications at Falmouth, they could be sent to London without any further risk of being lost at sea, either because of the weather or enemy action. Falmouth was home to the Post Office Packet Service, so the route to London was well established and the harbour was well suited to those vessels.