Just for fun, I fired up AutoCAD last night and drew two rims with 14mm and 19mm inside widths. On the 14mm rim I drew a tire of 25mm width that I will call the 25C. Also a 28mm width tire (28C).

Now, I assumed that the distance around the cross section of the tire from the inside of rim to inside of rim for the various width rims does not change for a given tire. Also that the tire takes the shape of an arc of a circle. (Ignoring possibly varying depth of thread and any forced molded shape. So what a quality high TPI tire with no tread at all would do. I also ignored the varying wrap around the top of the rim hook.

Looking at the 28C. Width for the 14mm rim is the spec'd 28mm. Height of that tire is 26.1mm.

Width for the same tire on the 19mm is 29.9mm or 1.9mm wider or 6.8% wider.
Height of the tire is 26.5mm or 0.4mm higher than the same tire on the 14mm rim. (1.9% higher.)

Now, running that 25C tire on the 19mm rim:

Width = 27.1mm or 0.9mm narrower than the 28C on the narrow rim.
Height = 23.1mm or a full 3.0mm lower than the 28C on the narrow rim


So, boosting rim width 5mm increases tire width ~2mm and height ~1/2mm. But dropping the tire one size decreases the width ~1mm and height a big 3mm.
(I don't claim this numbers represent any tire out there. But the trends should work for any manufacturer that is consistent in it's specs and QC.)