My understanding is that Moffat and colleagues went beyond merely introducing nonlocality. They made the nonlocal theory gauge invariant, and also used nonlocality as a clever regularization scheme. They also showed that the nonlocal theory does not necessarily violate macroscopic (classical) causality. I'm afraid that that's the best I can do by way of a "plain words" explanation; if you are willing to read the actual papers, perhaps the best is "Nonlocal regularizations of gauge theories" by Evens, Moffat, Kleppe and Woodard (PhysRevD 43:2, pp499-519 (1991)).

Dear Dr. Toth, thanks for your kind answer to my question. Your answer seems to imply that Yukawa only introduced nonlocality but that Moffat developed the theory much further than the former. Right? I would like to know how the latter was able to advance the theory further than the former. Did Moffat introduce some new assumption or use a new technique of calculation?

I don't think I'd characterize it that way. First, while Yukawa may have been the first (I am not sure) to systematically study nonlocality in a quantum field theory, the concept of nonlocality has been around for much longer. Second, Yukawa's nonlocality appears to have been motivated, in part, by the desire to introduce finite-size (as opposed to point-like) particles; indeed, his nonlocality explicitly makes use of a displacement vector, such that fields become functions of not just the coordinates but also this displacement. The type of nonlocality that Moffat studied had to do with operators that, effectively, contained an infinite number of derivatives. This has the effect of "smearing out" vertices, but the motivation here was not to create particles of finite size, but to come up with a novel scheme to avoid the need for renormalization; they also demonstrated that it is possible to create a gauge invariant nonlocal theory. So I'd argue that rather than one (Moffat's work) being a continuation of the other, the two represent quite different approaches that happen to have a common characteristic, namely nonlocality.

Thanks, Dr. Toth, for your second reply. I wrote the above inquiring comment because my original question was about the "relationship" (including the possible case of non-relationship) between the treatments of the two physicists while your first answer mentioned nothing about Yukawa's work without directly replying the question. Now your two replies together give me a satisfying answer to the original question.