On the Moon, 'recent' means anything after the formation of the Mare (the dark areas seen from Earth) which ended around 1 billion years ago.
The Moon is still seismically active as there are regular moonquakes which come in a variety of flavours. There are deep quakes which can go down to about 700km which seem to be associated with lunar tides, then there are the very shallow weak thermal 'quakes which are kicked off when at lunar dawn when temperatures skyrocket and rocks expand. In between there are shallow 'quakes down to about 30km which are caused by crustal movements. These tend to be bigger - IIRC the biggest measured by Apollo sensors was Richter 5.something. So rocks are definitely moving - why, well that's an interesting question. But it is only creating a tiny fraction of the seismic activity found here on Earth.
There's also some suggestion of intermittent volcanic activity or outgassing. Astronomers regularly report seeing glowing patches of light around certain craters which go by the name of Transient Lunar Phenomena:
Finally there are similar scarps on Mercury known as rupes. The accepted explanation is that Maercury's surface cooled relatively quickly and became rigid before the underlying Mantle and Core solidified. As the Core and Mantle cooled and contracted, the crust buckled along the rupes.
Nice image here:
The scarp is the dark line running almost vertically through the middle of the image cutting through the large crater. It's 2km high.