Most likely a .380 in a 9mm chamber would not fire at all. Most autoloader cartridges are rimless and headspace on the shoulder, in this case the leading edge of the case, so there would be too much slop in the fit for the firing pin to hit the primer unless you pointed it straight up! Even then a case rupture shouldn't harm a pistol built to withstand 50% greater chamber pressure but still not a good Idea.
I used to reload 9mm for my Browning Hi-Power using 80 gr. .380 bullets loaded to 1400 fps, it was quite spectacular.
I'm sitting here with a 9mm round (Winchester factory load) and a .380 round (PMC cheap commercial reload). I also have a micrometer. The .380 reads 0.368 at the base. The 9mm reads 0.3862.
I think the ejector on the wife's Sig held the .380 round in place even though it was 0.0182 inches smaller than the 9mm round.
/john