Well this is interesting..
I looked into this briefly to better understand the mechanics behind it, and I believe all that matters here is that the character walks (not pivot then walk and not dash) out of a specific amount of landing lag (specific to that character) with some momentum already carrying them in the direction they want to walk, then doing a specific action (that allows for sliding e.g. grab or shield) on a specific frame of the walk (different timing gives different length slides).
So with Fox for example, you don't need to do an aerial at all; the aerial just makes it so that he gets hard landing lag out of a SH (even when the aerial autocancels, so SH Uair and SH Bair will both work just fine). Otherwise, an empty FH (just for example) will also work just fine. Once you've guaranteed the hard landing lag and you've landed with momentum carrying you forward etc, it's all about the timing of (in this case) the grab. To give you an idea, grabbing on the first two frames of the walk didn't appear to have any real effect. Then from frame 3 of the walk Fox would get a notable extra slide, increasingly more notable with every frame, with frame 7 being the longest, and frame 8 taking it back to no slide.
[Also the shield slide thing works for Fox too.. just less pronounced than the grab slide.]
Bayo's work's slightly differently. You can't do it out of hard landing lag, instead it needs to be done out of a certain amount of aerial landing lag, with the Nair's 10 frames of landing lag being pretty much optimal. So e.g. you can just Nair at around the peak of a SH so that you land with Nair without having to hold A or anything; jut so long as you get the 10 frames of landing lag which can of course be made easier in some instances by holding A. Grabbing on frame 3 of the walk will give you a small slide, grabbing on frame 4 of the walk will give you a large slide, etc up to frame 7 giving the longest slide, while grabbing on frame 8 of the walk will result in no slide. Because of the differences in landing lags between Fox and Bayo in the two examples as indicated by the OP, obviously the execution timing for the grab after landing will differ even though they have the same frame requirements for interrupting their walk.
The difference between the two had me curious though. Why was it that Fox could use hard landing lag whereas Bayo required aerial landing lag?
I tested all of Bayo's aerials. On top of landing with Nair, it works for landing with Bair, Uair and Fair too, but to a lesser extent. With landing in Fair's landing lag, you need to grab on either frame 6 or 7 of the walk, otherwise it won't let you slide at all. And this makes sense I guess because you're stuck in landing lag for longer. The same seems to apply to Bair and Uair; it works, but you just have a slightly smaller window to grab in.
I believe that there must be a certain sweet-spot amount of lag which will differ for each character (that is, each character this actually applies to). Just enough lag to cause the walk to start up faster, but not too much so as to overlap too much with the window in which you can cancel the walk to get the slide.
Everyone should probably look into their own characters to see if they can find a sweet-spot amount of lag that is compatible with their walk. [Use frame skipping techniques as seen in the labbing thread in my signature to help you test accurately.]