I don't think post-processing can duplicate all perspective controls. I ask, sincerely, whether the use of front rise (to avoid converging verticals) can be duplicated successfully - I've seen it done, but it seems to be a distortion that looks different than actual use of front rise.
I wouldn't dare compare every different combinations of movements in LF camera and what they do to what is possible in Photoshop after a shot is taken with a regular lens in a DSLR. It's been a while since I played with my Toyo 4x5 and cracked open my Leslie Stroebel book (not that I was an expert in the first place.) However at first blush, given that various movements are nothing but manipulation of the projection of a 3D object onto a 2D ground glass by pushing and pulling the rays to make them fall at various places on film, I would think at least theoretically, digital post-processing should be able to get there by similarly pushing and pulling the pixels. Now granted, that when this kind of post-processing is done, there are new pixels created by interpolation and extrapolation which would deteriorate the quality of image somewhat. However with some of the newer higher pixel-count cameras, as well as better algorithms, this becomes less of an issue. The matter of fixing the converging lines is fairly straight forward:
https://helpx.adobe.com/photoshop/atv/cs6-tutorials/correcting-perspective.html
Nowadays I use something called the "perspective warp" tool that can do this (an much more!) with a few clicks.
One major difference I can think of here is that one has to take the picture with an idea of what kind of post-processing is going to be done (previsualization?) - parts of the image will get cropped out so one has to compensate by keeping extra space around. In a LF camera, you are able to see exactly what you are going to get by looking at the ground glass - something which a DSLR can not match at this point. It is still not clear to me whether or not the final corrected image is
identical to the one that may be obtained with in-camera correction. There may well be some artifacts and distortions as you allude in case of the former, rising not the least because of the skill level of the practitioner.
As another example, imagine a shelf in the kitchen that has a rack of spices and other items. You want to photograph this obliquely so that the rack is at a 45-degree angle, yet have all the spices in focus, from closest to furthest. You don't use f/1000 to achieve this focus, but merely use swing to shift the plane of focus using the Scheimpflug principle. Can post-processing do that? Again, I ask sincerely.
This can be achieved by doing "focus stacking." Some of the newer cameras can do this automatically with one press of the shutter. It takes several pictures focusing at the nearest, the farthest, and several points in-between in the scene. One would take these shots into Photoshop (or one of the other more dedicated apps) and combine them into one image where everything is in focus. The case of a spice rack that you describe may be one of the simpler ones to execute as there are no discontinuities in the image (like a foreground and a distant background.) Of course, like everything in digital there are limitations, artifacts and other complications to pay attention to. And you don't know if you did it right until you come home and fire up the computer whereas it is WYSIWYG in case of LF camera.