RADAR vs Radio? Well if RADAR is Radio Assisted Direction and Ranging (a classic definition), then a radalt ain't radar; it doesn't tell you in which direction the nearest ground is.
Radalts assume that the first return (ie the nearest terrain) is the number you want to know about.
This is important to remember. The beam of a radalt is quite wide, about 80 degrees, so that aircraft manoeuvres don't hinder the radalt's operation.
Whilst 4.2 to 4.4 GHz is a good band in which to operate a radalt, it's not perfect. Some terrains, particularly coniferous forests under certain circumstances, act like radar absorbent material at this frequency, and you don't get a strong enough return off the trees below for the radalt to detect. This leads to an incorrect height reading, perhaps giving you the distance to, say, the lake over yonder instead. Probably not an issue in an airliner, but a bit of a fright for a fighter pilot flying fast and low!
4.2-4.4 GHz works well because it is high enough so that smallish terrain features are noticeable, you can make a useful beam shape for a reasonable antenna size, and get good precision. But it's not so high that, say, a ploughed field, causes too much scatter and poor returns. A ploughed field when observed at this frequency is 'electrically flat' (it looks like a mirror), but a building is not.
If you look at the history of bands used for radalts you'll see that the band of choice has gone up as the capability of electronic components has improved, and then stuck at 4.3 GHz for the past 30 years or so. Ground observation satellites are all up at 15GHz or so because at this frequency they can 'see' the furrows in a ploughed field, etc.
Most radalt waveforms (they're FMCW, usually a variable rate linear chirp) track the ground. They have good precision at low altitude, and less precision higher up. When first switched on they go through a ground acquisition search, and then reduce the chirp rate as the aircraft altitude increases, 'tracking' the ground. The tracking is rate limited, but it's quick, far quicker than your plane can change height.
So it all boils down to:
1) a radalt tells you the distance to the nearest ground, which isn't necessarily directly below you. RA approaching Kai Tak must have looked interesting! Over your undulating hills it's probably going to be to the ground directly below, depending on the exact geometry of beam width, terrain shape, etc.
2) that is rate limited by the ground tracking done by Radalts. Ultimately the needle on the gauge won't move fast enough to react to every single building passing by below, but it's quicker than your plane can manoeuvre.
Depending on the application some secondary rate limiting may be done. Radalts in hydrofoils need to track average wave height, not instantaneous wave height, to give a smooth ride.