So I watched this video talking about now slices, and how it seems that across vast distances of space, movement can affect what is actually the now of places far away.

https://www.youtube.com/watch?v=

This seems to be in direct conflict with what I've heard about quantum entanglement.

Let's say we move an entangled particle across such a distance. If we start moving away with our particle at very fast pace, the entangled particle on the other side will be affected in the past, where as if we start moving towards it, it will be affected in the future. But from what I've heard, entangled particles change "simultaneously" without regard to which "now slice" is happening.

Am I describing this correctly? Is anybody researching how these two phenomenon relate?

There is no such thing as a global "now". Or, perhaps more accurately, there are many perfectly-valid slices of the universe that can be thought of as having equal time values. What's incorrect here is your understanding of quantum entanglement. But that's understandable, as quantum entanglement is a very strange effect that is frequently described incorrectly.

Here's a (hopefully) better way to think about it: quantum entanglement is, fundamentally, about consistency. If we have a quantum-mechanical system that splits into two particles, one whose spin is always opposite the other's, then any measurement of particle A's spin will be opposite of particle B's spin. That is, if I measure spin "up" on my particle A, and particle B travels towards you, then you will necessarily measure spin "down" on particle B. It does not matter if we measure the particles at the same time, or one before the other. The timing is irrelevant. The only thing that is relevant is that the two particles have spins that are consistent.