The Iron dome analogy may not work here - While it is a VERY potent system - no operator of the S-400 is churning out all of its missiles.
There are firing patterns that can be manual and automated - but due to the cost of the missile its generally a modified program/manual that is run unless the operator/system has determined that it is the target.
Lets say the Indians decided to put a battery at Adampur(which puts it at risk from a lot of other things but for discussions sake). This lets it cover a large chunk of Pk airspace right up until Sargodha and secures most of its key bases in Kashmir too. I have given it protection via a Tungska and a QR-SAM Battery(in this case represented by a ground launched ASRAAM)
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A fatah is placed at Rahwali (in this case I am using a CM-400 ground launched simulated) that lets it just touch the S-400:
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When launched the "Fatah" is picked up by the combined Indian ADGE around 85nm from its intended target.
Technically while they are in the range of the S-400 missiles - based on various factors(curvature, target size, insufficient reflection from illuminators etc) the S-400 cannot engage it. I have set the S-400 to fire all missiles initially(not very realistic)
It is not until 34.5nm when the Fatah is in its terminal dive that the S-400's fire. It is overkill so the 2nd missiles take them out.
Now lets change the scenario to something more plausible -
the S-400 will fire 4 missiles at each target - from different batteries.
However, there are now 3 shooters - 2 Fatahs - one each at Rahwali and one south near Pattoki
And JF-17s firing off CM-400s from around Mangla -and the targeting sequence will be that the ground launched systems fire just a bit before the other CM-400s pass overhead.
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5 out of 6 were shot down - but one made it through and the S-400 could not re-engage again - neither the Tungska nor the QR-SAM shot anything at the incoming missiles and the S-400 was hit.
Now here is the fun part - I ran this a 6 times in this scenario and in 4 cases the S-400 is able to take all of them out - 2 time it missed but one CM-400 malfunctioned out of 2 remaining so only 1 hit.