Changing airports is almost never something you planned. Düsseldorf Airport to Frankfurt-Hahn Airport is 226 km and about 2h 30, and a car covers it in one leg with no change and no ticket barrier between you and your suitcase.
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 terminal door, not the car park. 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 Stuttgart Airport, priced the same way.
Luggage stays in the car for the whole 2h 30, which is the single biggest practical difference from any rail route: nothing is lifted onto a train at Düsseldorf Airport, off it again at the interchange, or onto a second one for the run into Frankfurt-Hahn 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.