Most of the time entanglement is described using some instantaneous classical measurement from an apparatus, but it is possible to describe it between two quantum systems, with two systems $O$ (for the observer) and $S$ undergoing the following process
$$\vert \psi^O \rangle \otimes\vert \psi^S \rangle \to \sum_\alpha a_\alpha \vert \psi^O_\alpha \rangle \otimes \vert \phi_\alpha \rangle$$
for a bunch of measurement with value $\alpha$ and probability $a_\alpha^2$.
But is there a toy model to show this process actually happening, where two particles, originally just a tensor product of the free states, evolve into entangled states, ideally without using some instantaneous interaction like a quantum gate? Also does this conflict with the fact that for long range interactions, particles are always interacting with each other?