If a random variable follows a random walk, is the expected change in its value correlated with its prior changes?
No, by definition.
A random walk is just that: random!
Every step of the way is completely independent of anything that has happened before. The variable either goes positively or negatively by some unit, every single time.
Therefore, by definition of a random walk: no, the expected change in value of a random variable following a random walk is not correlated with its prior changes.