In Michigan, we have a mix of dedicated and non-dedicated Thruway connections, and I’d imagine this is probably the case in many other states/regions as well. The previously mentioned bus to Toledo is an example of a dedicated Thruway connection - it isn’t an Amtrak-branded bus (like the Amtrak California buses), but it exclusively serves Amtrak passengers connecting to the Capitol/Lake Shore Limited from Michigan. The return trip to Michigan will wait for both trains before heading back - I’m not sure what happens if one is severely delayed, but I think they may send a second bus, as it would be awful to have to wait multiple hours in TOL. Eastbound LSL passengers already have to wait hours in TOL due to use of a single bus for both trains, though they’re warned of this by the schedule in advance (hence why I avoid the eastbound LSL at all costs...)
In addition to this (and perhaps other dedicated services I’m unaware of), Michigan also offers some Greyhound service as Thruway connections. One that was fairly-recently added was Greyhound’s overnight Chicago-Detroit run. I’d almost rather they didn’t add that one - assume it was probably added to serve all the missed connections from western LD trains, and they’re probably using this to avoid putting Michigan passengers up in Chicago hotels. Personally, I’d rather stay in a Chicago hotel and take the Wolverine the next day than take overnight Greyhound arriving at 5am, but I get the cost reasons for doing that. Though sticking people on overnight Greyhound could also backfire by creating more Amtrak “never again” horror stories...
Most of the Indian Trails bus network in Michigan/Illinois/Wisconsin is also available. Many (though not all) of their stops not at train stations have Amtrak station codes. Their buses aren’t necessarily timed with Amtrak or even Greyhound (in fact, I know some buses that are scheduled to just barely miss connecting), something that is incredibly annoying since Ann Arbor has no regular Indian Trails buses (they do operate the AirRide/Detroit Connector, but that doesn’t connect with mainline service). However, their buses are offered as connections for one-way/round-trip Amtrak itineraries when timing would allow for an Amtrak-legal connection. Which raises the question - how does Amtrak accommodate a missed connection booked one-way/round-trip to/from a bus, particularly when the next trip is the next day? I suspect they don’t have hotel arrangements in places like Kalamazoo (where you can connect to Indian Trails), though leaving people on their own with no accommodations in Kalamazoo for a day would probably result in outraged customers.