All rocket-type missiles of any size are very fast by virtue of form factor and design. Even the first such missile - German V2 - was essentially hypersonic. All modern surface-surface missiles are hypersonic at portions of their trajectories.
Kinzhal is a surface-surface missile that is adapted for air-launch. It gets a Mach1+ assist from the Mig-31 and climbs fast to tens of km, spending most of it's flight in very sparse atmosphere where drag is very low and it's relatively easy to fly very fast. Every space going rocket gets to Mach20+.
However, once it gets close to the target it enters a deep vertical dive and falls to Earth inertially - it's RM has long since burned out. As it enters the low atmosphere, it dramatically slows down - like everything that falls to Earth from space. In the intercept battle space, Kinzhal is probably about Mach4.
Patriot PAC-2 and PAC-3 are at similar speeds at intercept. If Kinzhal maneuvers near the target, it's not a lot - large maneuvers at Mach4 in dense atmosphere generate large g loads and take you off course quickly - and you no longer have time to get back on track.
Interceptor missiles are typically made to achieve high maneuver g factors - and for a variety of technical reasons, surface attack missiles are not. One of them is the intended target - Kinzhal carries a heavy warhead to destroy a large building, while PAC-2 has a much smaller warhead to destroy an airplane or a missile. PAC-3 has no warhead at all (it does carry solid rods along its body that are pushed off during intercept to enhance the lethal diameter), so it can maneuver faster than Kinzhal. And the interceptor's job is easier, since Kinzhal is falling vertically from space - *all* the interceptor have to do is position itself in Kinzhal's path. At Mach8 total combined speed, any contact with the PAC-3 body or the rods will not only make large holes, but will throw the Kinzhal off the stable near ballistic flight into an uncontrolled tumble, rapid deceleration and immediate structural breakup.
However, since it's in a vertical fall over the target anyway, pieces of the Kinzhal will fall down slowly under gravity, breaking cars, setting things on fire and generally creating a nuisance in the city environment. But it's far less dangerous that a half kiloton fully fused high explosive warhead that Kinzhal carries. Note there were no deaths reported after the fierce engagement in the skies.
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