For example: was it caused by poor grading/drainage, tree roots, post-tension cable failure, water leak, poor moisture management (i.e. no sprinkler)? 

Your worst enemy here is moisture, so yes make sure your HVAC system is working properly and make it clear to the tenant that they notify you of any dramatic changes to the floor or noticeable water damage.

It is important for me that the whole system is checked properly, and I can be sure there will be no surprises like leaking valves or moisture in the basement.

I think if the city guy came by, he'd go nuts and the local foundation guy knows the rule.If the building is just resting on dirt w/o moisture barrier, mucho problemas.

FedPac breakers2) Moisture invasion like bad roofs3) Tenant safety like steps you can step through or floppy railingsMay mean no renew or higher rates.  

What does damage caused by shifting look like vs moisture damage?  

Foundation SurfaceThe exterior surface of the foundation is Parged (covered with a thin coat of a cementitious material for refinement of the surface).Fill and seal all of the cracks in the foundation to reduce moisture penetration and relateddamages.Animal hole noted along the foundation Cracked/damaged sections of the parge coat noted on the exterior foundation wall; recommend repair to help prevent moisture entry and additional damages.

If its good congrats super easy fix, still buy gfci outlets for wherever theres potential excessive moisture ex.

You'll know in an instant whether there's moisture or heavy mold. 

As the temp decreases, the dew point rises (assuming moisture in the air is trapped in the house).