TLDR version: Don’t do it. Stop thinking about it.
Longer version, if you want to know why:
Disclaimer: The information below is theoretical only. It is a fundamental safety rule to only use ammunition for which the gun was designed. Using the wrong ammunition is a good way to ruin a gun, get injured, and possibly die. There are a small number of guns that are designed to handle multiple chamberings and the manufacturer will indicate when these are safe. To my knowledge these are:
.410 shotgun shells may be used in a few .45 colt revolvers and derringers designed to allow this. MOST .45 colt revolvers only handle .45 colt however.
.45 colt rounds may be safely used in .434 Cassul guns.
.44 special rounds (rare) may be used in .44 magnum guns.
.38 special rounds may be used in .357 magnum guns.
In the last three cases, the more powerful round is a “magnumized” version of the less powerful one. The cases of the higher powered round are always a bit longer so that they will not fit in guns not designed to handle them. Using the lower powered rounds may not be without problems, but it is not dangerous.
In addition there are a few guns with interchangeable cylinders are barrels that allow additional calibers to be used with no issues. .22 LR and .22 WRM and .45LC ancd .45ACP are examples that Ruger currently produces revolvers with multiple cylinders.
First off, a .380 (or 9mm) bullet is a couple thou smaller than a .38 (.355 vs .357) it will not cause excessive pressure, but accuracy may be poor. The .380 bullet might barely engage the rifling, and perhaps not at all if it is very shallow, and it may not seal well, regardless. I believe at least one manufacturer (S&W or Taurus?? heck, maybe it was Ruger?) DID offer a revolver with 9mm and .38spl interchangeable cylinders, so apparently the bullet diameter is close enough for jazz?
Next, as has been mentioned, There is no rim on a 380 round, thus nothing to hold it in the proper position for the firing pin. If you pointed the revolver upward, there might be enough inertia from the round to allow the percussion of the firing pin to fire it. Then the REAL trouble starts:
The firing pin will drive the case forward before the powder really gets burning. Eventually the powder lights up and the pressure builds, forcing the walls of the case against the walls of the chamber. This is what seals the breech of all cartridge based arms, even shotguns where lower pressures allow paper or plastic hulls to do the job. With the walls of the case press tightly to the chamber walls, the rear of the case (the “head”) is driven rearward until stopped by whatever closes the rear of the chamber…bolt, slide, revolver frame, falling or rolling block, etc. I’ll just call it the breech block. This stretches the case in the area just forward of the head. If the distance from the rear of the case to the breech block (“headspace”) is too much, then the case will tear (“rupture”) and high pressure gasses will be vented rearward. This is bad, but occasionally happens, so firearms are usually built to protect the shooter from the problem. Many guns protect right-handed shooters, but expose left-handed shooters, and many of the few left-handed guns do the opposite. A true ambidextrous firearm must vent gas from a ruptured case or primer in a way that protects both righties and lefties.
Note that cases get stretched slightly even with correct headspace, as there must be a little headspace to allow a maximum tolerance length round to be fired without jamming. If a case is reloaded many times, it will eventually rupture due to this stretching. This is something experienced reloaders are aware of and alert for.
Anyway, the point is that the head space must not be excessive, or you can expect the case to rupture, and if you manage to get a .38 to fire a .380 round, it is very likely, virtually certain in fact, that the case will rupture, venting hot gas, possibly injuring the shooter, the gun, and almost certainly jamming it to the point that extensive disassembly and inspection will be required to return it to service. It might cut and/or burn your hand, might blind you, but you will most likely live.
Revolvers typically use rimmed cases to control headspace. The thickness of the rim and the fit of the cylinder to the frame determine the headspace. This is what allows some revolvers to fire different length cases. It also simplifies the ejector for double action revolvers.
Rimmed cases often cause feeding trouble in stacked magazines. So rimless cases were a solution. There is an extractor groove, but the “rim” behind the groove does not extend beyond the body of the case. 9mm and .45 ACP were initially intended to approximate the ballistics of .38 spl and .45 colt in a rimless case, with smokeless powder loads.
Since rimless cases lack a rim to control headspace, some other solution is required. Bottlenecked cases (mostly used in rifles) headspace from the shoulder where the case diameter transitions. Straight walled cases (like 9mm, 380, .45 ACP) headspace off the case mouth.
The few revolvers that work with rimless ammunition have cylinders with chambers that provide a precise step for the case mouth to control headspace. They also use either half or full-moon clips to allow the ejector to work. The clips are not required for headspacing or safe operation. In a pinch the revolver may be used without the clips, and cases ejected one-at-a-time using a pencil or similar. Revolvers that fire rimless ammo are the exception, but not exactly rare. Many S&W .45 ACP revolvers were issued during and for a while after WW1, because production of the 1911 automatic was not sufficient to supply demand. S&W revolvers helped fill that gap, while still using the same ammo.