The reason why "Which incoming particle is identical with which outgoing particle?" is a meaningless question is that it involves a false assumption — the assumption that what really happens is either of these alternatives:

Elastic scattering of two particles of the same type

The question arises because we assume that initially there are two things (one moving northward and one moving southward), that in the end there are two things (one moving eastward and one moving westward), and that each thing persists as an individual that remains identical with itself and distinct from every other individual.

One way to avoid this meaningless question is to assume, instead, that initially there is one thing moving both northward and southward, and that in the end there is one thing moving both eastward and westward. Now we cannot ask: which is which? Or else, we now have a reason why this question is meaningless.

And in the meantime? Since no property is possessed unless its possession is indicated — by an actual event or state of affairs, for nothing else is capable of indicating anything — and since nothing is indicated as far as the system's properties in the meantime are concerned, there is one thing. Period.

We know there aren't two things, but how do we know there is one thing, apart from it being a way of avoiding a meaningless question?

The scattering event we discussed was assumed to be elastic. This means that we ignored that particles can be created and/or annihilated. In other words, we assumed that the number of components of a (closed) system is a conserved quantity — every time it is measured, the same value is obtained. This assumption is valid in non-relativistic quantum physics, which is an approximation to the correct theory, which is relativistic. It allows us to think of the components as constituting the system.

In the correct, relativistic theory, particles can swap their identities. Here are two of the (infinitely many) alternatives that contribute to the calculation of the probability of a specific inelastic scattering event:

Inelastic scattering of a proton and a neutron

Even though we now have two distinguishable particles coming in (one neutron and one proton) and two distinguishable particles going out (one neutron and one proton), we still cannot say which incoming particle is identical with which outgoing particle. Since the identity tags "proton" and "neutron" can be swapped through the exchange of an electron and a neutrino (the nuclear reaction responsible for β-decay), the distinction we make between the two alternatives is another distinction that Nature does not make.

In relativistic quantum physics, a system's number of components is an observable like any other: it can change, and it has a value only if, only when, and only to the extent that a value is indicated. This makes it impossible to think of the components of a system as constituting the system. The system as a whole comes first. It exists in an anterior logical relation to its "parts." The number of "components" that "make up" the system is simply one of its properties.

But then our theoretical division of the world into separate physical systems, or into one physical system and "everything else," is ultimately a fiction. At bottom there is only one system, of which we ought to think an intrinsically undivided whole. The number of its parts and their properties are its properties. Rather than being constituted by "parts," this constitutes the universe and each of the apparently separate things that exist in it. In our elastic scattering event, this is what moves both northward and southward initially and both eastward and westward in the end.