Dubrovnik to Split Airport

Flixbus seems to to the trip.

The buses are pretty much hourly. There should be a shuttle bus from airport to bus station (check DBV airport website) , where you’d change to intercity bus (google “Dubrovnik bus station” and that should bring up the schedules.

There’s no train between Dubrovnik and Split. While there are buses, schedules can be tricky and they may not be the most comfortable, especially if you’re carrying luggage or arriving at odd hours.

A private transfer can be a great alternative. Some local drivers offer door-to-door service between the cities, often with optional sightseeing stops. It’s especially convenient if you're flying into Dubrovnik and want to go straight to your accommodation in Split without dealing with bus logistics.

One option to consider is Ivan Ćosić of MC-Travel Croatia. Based locally, offers new, comfortable vans, and can be found on Facebook if you'd like to inquire directly.

Flixbus should have you covered. They have a nice app too. Alternatively, you might check Croatia Airlines for a faster, though undoubtedly more expensive alternative.

I don't believe there is a direct flight--it's roughly four hours on the bus. If you don't want to do that, there are catamarans (which take longer, cost more, but schedules are here in case you are returning via the sea: https://www.ferrycroatia.com/en/dubrovnik-split/.)
Airport shuttle to bus station:
https://platanus.hr/shuttle-bus/
Bus schedules:
https://autobusni-kolodvor-dubrovnik.com/en