World background a finite distance from the origin? Blender treats the world background as infinitely far away, meaning that moving the background as seen through a camera moving through the space does not scale. 

Is it possible to have Blender treat the background like it is a finite distance from the origin, producing the sensation that the camera is actually moving through the space (relative to the background image)?

My intuition is that I'll have to map my texture onto a large sphere/box, but I'm not sure how this affects background lighting.
Overview:

You can use your HDRi map as a Emission color on the Sphere (big one, mine is above 100m in diameter). Set Image Texture projection to Sphere as well.



Differences:

Using this method you will get more noise as for any mesh light in Blender. Everything else like reflection, shadows etc. should be the same.

Environment map (HDRi):



HDRi mapped to the Sphere: