This is an interesting, though I think a bit impractical, way to adjust a bracelet or strap on a watch. Rather than having a fine adjust on a clasp, as is most common, this puts the fine adjust on the lugs, effectively.
In the various embodiments of the invention, at least one set of lugs on case can be moved in or out slightly via a screw or slide. This would allow for minor adjustments to the strap over the course of a day.
Some, perhaps obvious, potential downsides of this would include that it makes design a little harder when there isn't a fixed lug-to-lug dimension. And the watch would have to look good with a variety of spacings. It would also make it practically impossible to have an end link in a bracelet that was flush against the case.
Finally, for safety and function, the slide would have to be integral to the case, which would likely thicken the case of the watch pretty dramatically, though with a small movement in a larger case this issue could be minimized.