I have this reply now from the Beamdog support staff, so basically you have two modes: transport-picks-up-unit and unit-gets-on-transport. The former is a bit slower to use but the only way to get things to work when the movement is more complex, such as if a transport must pick up a unit outside its starting area or be very explicit in what sea zones it moves to:
From Beamdog
Transport -> Unit Loading (Original, Classic method)
Select Transport Stack (left-click on Transport Stack you want to move)
Load a unit in an adjacent region if desired (right-click on Unit you wish to load)
Move Transport to Region you want to move the Transport to (right-click on target region (blue highlight))
Load a unit in an adjacent region if desired (right-click on Unit you wish to load)
Optional: Move Transport to Region you want to move the Transport to (right-click on target region (blue highlight))
Unload units into target region (right-click on territory you wish to unload the units into. Transport will move to the adjacent region and automatically unload the cargo into the desired region)
Unit -> Transport Loading
Select Unit you wish to load into Transport (Left Click on unit stack)
Move the Unit into the Sea Zone with a Transport (right-click on target sea zone)
Select the Transport Stack in the adjacent sea zone (left click on Transport Stack) OR click on Destination Marker for loading to open the Dossier to select the Transport.
Optional: Open Dossier, confirm which Transport is selected by observing the glow effect around the desired Transport. To select the desired Transport, left-click on the Transport you wish to move)
Move Transport to Region you want to move the Transport to (right-click on target region (blue highlight))
Select Unit you wish to load into Transport (Left Click on unit stack)
Move the Unit into the Sea Zone with a Transport (right-click on target sea zone)
Select the Transport Stack in the adjacent sea zone (left click on Transport Stack) OR click on Destination Marker for loading to open the Dossier to select the Transport.
Optional: Open Dossier, confirm which Transport is selected by observing the glow effect around the desired Transport. To select the desired Transport, left-click on the Transport you wish to move)
Optional: Move Transport to Region you want to move the Transport to (right-click on target region (blue highlight))
Unload units into target region (right-click on territory you wish to unload the units into. Transport will move to the adjacent region and automatically unload the cargo into the desired region)