More on topic, I think our existing theories of reality tell us that the future probably affects the past, or at very least, is predetermined. If you know the solution to a differential equation, you know the state at any point in time. There's no going off the rails, so to speak. And if the system is constrained in any dimension (i.e. it's nonholonomic) then the state at any given moment relies on the full trajectory up to that point, as opposed to just the previous state. But you can flip that on its head and say that the future state determines the present trajectory. E.g. to turn a bicycle left, you have to first turn right.
Our brains like to process events in a linear sequence, from past to present to future, but this view of time might not fully capture how everything in the universe is interconnected. It's possible that our language just can't handle these complex concepts.
So, I don't think anything mentioned in this thread contradicts the idea that "time always moves in one direction" and "entropy always increases".
I'll expand here how if the arrow of time actually doesn't exist, the future could change the past without breaking our - current - understanding of physics (tragiclly because we will probably have maybe hundred of years till we change our minds about what are the properties of the universe).
In a range or probabilities of one event, one of the probabilities will always be the one is going to actually happen. Closer to the event to occur the most certain you can be about which one of those, could be the one that will happen.
Here's the catch, you have to change your mind a bit, in our current understanding of physics, the information of the future event which has not happen yet, already exists here in the past for that event.
Now, if you know - and you know - the outcome of an event, you most certainly will be able to choose if you go along that path, or you choose to change the future event to something else. Whoalá, the future changed the past.
Brains of earth continously do this, every second of our entire lives, animal brains do this too, almost all life on Earth has some neurological, physiological, chemical predictive mechanism in place.
An example, you're about to walk across a street, you see a car coming, now your brain "predicts" you'll be hit by that car if you just keep walking, you stop walking to avoid certain death, done, the future changed the past. The information - really close to 100% of probability of death - extracted by the brain from the sheer reality of the universe, allowed you to survive.
And no current understanding of physics got harmed anywhere along the way of the future changing the past.
So no, it's not at all incompatible with retro-causality. When you think about retro-causality, the idea that time "moves" is completely superfluous. Time is probably not even a real thing.