I don't know for sure, obviously. But it could take many forms. Presumably it will be leaving from the caravanserai and take X time, possibly with events or encounters as a set piece. Then once there, there's a waystone and you just roam on back. Or, barring that, once Astrid gets back home, she informs the higher-ups. She can't roam the Way Between curently, but she may have a stone like yours atturned to Hawkethorne (or someone there does) and, at the appropriate time for the game, they come to meet the Champ in Hawkethorne and they open the way and lead them to Tychris, which allows the Champ to attune their own waystone to that end and come back, thus spending only a short time in travel (maybe a day or two rather than weeks).
Or maybe like you said, a carriage. Maybe Baroness Carmen has had her old, black horse-drawn carriage brought from the stablehouse., drawn by a team of silent, black horses.. maybe it doesn't even use horses, just misty shadow horses... and the party rides there with Barnaphous.
Plenty of reason to go to Tychris, it's Nereva's patron city, she's the goddess of knowledge and secrets. The likely have libraries. Plenty of places to look for the location of artifacts for how the older empires and such dealt with demons (like the artifact Tollus was seeking in Sorra's spot, where you got her spear). Or ways to open portals since getting Cait's sister from another realm is still a reason or maybe they think they need to retrieve something from the demons' home plane. Or just as an ambassador to get more allies to fight the demon army.