How does entanglement work independent of time?

The title of the article and the abstract of the article both say it is entanglement swapping and you described entanglement swapping, so it seems like entanglement swapping.

The authors also mention that one interpretation is the measuring the first particle on the first entangled pair steers the dynamics of the second pair. In fact, this is normal, if you have two maximally entangled particles and you measure one of them, you have now placed the other one into a definite state.

So effectively you have a photon in a definite polarization state and then you do a projection (measurement) onto a bell state. The result of the bell state measurement is to find out which bell state it is now in. And it performs an entanglement swap.

If the first pair were entangled to have the same polarization as was the second, then when you do a measurement of one of each pair into a bell state you entangle the other two.

For instance you could get a bell state where the two particles measured into the bell state have the same polarization so now the other two have the same as each other. If instead you get them in a bell state where they have opposing polarization then the other two now have opposite polarizations.

This should sound entirely undeep. And it isn't deep, it is a regular entanglement swap, and even the authors say that you can interpret it as the measurement of the first particle influencing the latter results. It isn't like the polarization measurement result on the first particle determined everything, it doesn't tell us which bell state you get, but if you did the experiment with matter then there is a well defined probability current and which part of the beam ends up where is influenced by "irrelevant" factors like if you adjust the spacing for the bell measurement to give an extra half wavelength to one of the paths to adjust which detector corresponds to which bell state.

And all the interesting quantum correlations come not from a single measurement, but from correlations of different kinds of measurements on particles 1 and 4.