When a unit is selected, its stats are shown in the lower right part of the screen:
Recently I noticed that the small icons next to the combat strength and the movement points can be clicked and behave like toggle buttons:
However there is no tooltip and I didn’t notice any action triggered by these buttons.
Do they have any function? If so, what do they do?
As of the Spring 2017 update, and perhaps some time before that, this is no longer an issue: the buttons have been changed to icons, clicking on them no longer changes their appearance, and the tooltip text for the question mark is the same as for the image of the unit itself on the left (unit name and hit points).
There is no tooltip for the movement icon; if you leave your pointer over it, you’ll eventually get the tooltip for the underlying hex.