No trailers or modulars, but an actual stick built home.
I know this was common years ago to do, but not sure about now.
Anyway, I know were a 30's or 40's era home is on an acre or so of land that was moved from a few miles away.
Decent price, but needs siding.
Just wonder what I should look for when looking under it.

Water damage, obvious structural issues, floor joist on center, etc..
Today homes can be successfully moved to a sound and solid foundation with no issues if done correctly. Not sure what stage you are looking at it(already moved, on way,just moved, etc.)
I would also pay particular attention to the interior walls and if they have shifted excessively(plaster or sheetrock cracks and splits), level floors and walls.

Make sure they remove the wheels

Not a home owner but I grew up in a couple of old houses.

Anything built in the 30's will have whacky electrical, whacky plumbing, poor insulation, and renovations can be very difficult as differences in codes and lumber specifications make it tough for construction guys to plan the renovation without ripping walls down to framing for "lets have a look-see at what we're working with here"

Check the flashing around roof vents and chimney. The moved home I was in had water damage because cracks formed around the pipes and the flashing pulled away. Water followed the pipes into the house

My parents house was moved.
Built in the 20's
Moved in the early 80's

Stucco has pretty bad cracks.  Requires sealing and repainting every 5-7 years or so.
All doorways have cracks in the plaster at the top corners.
Main floor has a bit of a ledge between living room and dining room, mostly fixed with a couple steel pillars added in the basement.

Other than those things the house is in good shape.