The 268 km between Frankfurt-Hahn Airport and Stuttgart Airport take roughly 2h 48 by road. Booked as a transfer it is one vehicle from one terminal kerb to the other, at a price agreed in advance — which matters more than usual on a day that has already gone wrong.
The pickup is driven by your inbound flight number: the driver watches the actual landing time and waits 60 minutes free after touchdown, which is what a diverted or delayed arrival needs. At the far end the drop-off is at the correct terminal for your departure — Stuttgart Airport has 4 and the wrong one costs a shuttle ride you do not have time for. Tell us the departing flight at booking and the schedule is worked backwards from its check-in deadline rather than forwards from a convenient departure. We run the same journey to Munich Airport, priced the same way.
Luggage stays in the car for the whole 2h 48, which is the single biggest practical difference from any rail route: nothing is lifted onto a train at Frankfurt-Hahn Airport, off it again at the interchange, or onto a second one for the run into Stuttgart Airport. If the airline has rebooked you and is reimbursing ground transport, the invoice shows the route, the date and the fare in a form an airline will accept.
Booking this for somebody else
A large share of these journeys are arranged by someone who is not travelling: a colleague at a desk in another timezone, a family member watching a flight tracker, or airline ground staff working through a queue of rebooked passengers. The booking is built for that. What it needs is the passenger's name, the two flight numbers, and one mobile that will actually be answered — which is usually the booker's, not the passenger's, since the passenger is in the air for most of the window that matters. Confirmation goes to both addresses, and the invoice goes to whoever is paying, which is frequently a third party again.