When something like this is launched a few thousand kilometres, how hard is it to be shot down?
Well given the speeds of (hypersonic) re-entry involved with ballistics, they are lot harder to shoot down than other aerial targets.
A whole new class of sensors and sensor networks are required along with the computing power (for calculating firing solution interception) and missile systems for final interception at whichever stage of trajectory that is optimal and/or possible.
Essentially a new level of air defence system (involving space as well).
There are various design challenges in all of this.
Take just one, sensor networks (over the horizon radars etc) are aided by the ionosphere when it comes to early warning (by essentially reflecting certain high frequency radar waveforms between the ground/sea and the ionosphere)....but are also encumbered by it for ground based sensing needs when it comes to detection+tracking above the ionosphere (say for activation of the engagement radars as early as possible)...and then the need to shift and locate some of these sensors in space themselves.
Then the latter's final discernment of real incoming target versus onboard ballistic penetration aids like chaff and decoys etc....and also the ability of non-ballistic trajectories (depressed using thrusters, glide and so on) that may be employed by the incoming target.
Interceptor missiles also need to be resistant to penetration aid jammers and so on.
Addressing all these issues needs lot of research, resources spent and continued re-investment.
These make what is known as ballistic missile defence.
The thread concerning India's approach is here:
MOD NOTE: All developments related to the Indian BMD and ASAT program to be compiled here. @Nilgiri write a thread a few days back. Don't know if you have seen it :
defencehub.live
This summary here of the various systems involved gives a bit more that you can look more into as to your interest:
Warning system - Alerts, Detection, Prevention: Radar and identification friend or foe (IFF) equipment constitute the forward elements of complex systems that have appeared throughout the world. Examples include the semiautomatic ground environment (SAGE), augmented by a mobile backup intercept...
www.britannica.com