There are many factors at play.
1) the angle at which things appear to move in the sky, relative to your meridian, is always around 15 degrees per hour (360 in 24 hours), except for the Moon which is NORMALLY slower (360 in 24.87 hours = 14.5 degrees per hour). However, the Moon's orbit is elliptical and close to us, so that there are parts of the orbit where the Moon goes a bit faster and parts where it is slower. The difference could amount to 1/4 of a degree per hour.
2) The rate at which things appear to move relative to the HORIZON will depend on the angle at which the body sets. This, in turn, depends on the latitude of the observer and on the declination of the object.
If you are at the equator, then everything sets straight down (vertically).
If you are at a temperate latitude, then things set an angle close to the complement of your latitude (90 - LAT).
If in addition, the body has a strong declination same name as your latitude, the angle will be shallower and the body will seem to take forever to set. At the limit, if you latitude is the same as the co-declination (in other words, if LAT + DEC ≥ 90) then the body never sets (midnight sun, for example).
On the other hand, if the body's declination is opposite the latitude, the angle is steeper and the rate of setting appears faster than usual.
The Moon's declination changes rapidly, covering a range of +28 to -28 in two weeks, and -28 to +28 in the following two weeks. Therefore, things change rapidly from night to night (or day to day because one can observe moonset during daylight hours this week).
3) Slope of the horizon.
Calculations, in astronomy, rely on a straight horizon (for example, the theoretical horizon, located at 90 degrees from the zenith, in all directions).
If you live in northern latitude, with the western horizon sloping upwards towards the right, the same direction as the change of azimuth of setting objects, then the observed rate of setting will be faster than the calculated rate.
4) refraction.
Normally, there is more refraction (due to the air) as one gets closer to the horizon. Since this refraction acts upwards, it tends to slow the apparent rate of setting (the closer to the horizon, the higher the image is refracted).
However, this is based on a model atmosphere where the pressure and temperature change gradually with altitude.
If there happens to be an air layer inversion (warm air layer below cold air, for example), this could change the behaviour of the refraction, causing the "image" to set faster than usual.
5) the false horizon.
Some atmospheric troughs cause a localized middle-layer cloud bank (altostratus, for example), sufficiently thick to block off the light of the Moon. On Earth, in temperate latitudes, such systems usually come from the West. If there is such a system to the West of you and far enough to put the edge of the cloud bank just above your real horizon, you may perceive the cloud edge as the horizon.
If the system is coming towards your (from the west, as is usually the case), the take horizon would be climbing slowly. If you do not notice that the horizon is actually a cloud bank creeping upwards above your real horizon, you will think that the Moon is setting faster than usual.
And if you combine many of these effects, then you feel the Moon is setting exceedingly fast.
I witnessed some of these, plus the effect of moving eastward at high speed (which makes the objects in the sky appear to be faster in their apparent westward movement). Since your question does not mention you being on a vehicle, I did not use it. It would be number 6 on the above list.
6) If you are on a plane flying east at 480 knots (roughly 900 km/h) at latitude 60 (usual polar route from North America to Europe, for example), then the plane adds a further 1/4 degree per hour to the apparent westward movement. Not much for eyeball observations, but we do notice it with sextant observations.