I study materials science so this is right up my alley. I can't comment on how it affects the final product but I could give some physical insight.
What exactly is carbon migration during the forging process?
Luftmensch gave a pretty good description here. It's where carbon atoms in one region of a steel move to another part of the steel. It shouldn't happen significantly in monosteel knives, since it requires a difference in carbon concentration to happen. Monosteel knives should have uniform(ish) composition and concentration throughout the bar.
However, consider a three-layer knife. Each layer has a unique composition. Say the center layer is carbon steel and the outer layers are iron. Take a simple carbon steel, 1095 carbon steel. It is 0.95% carbon and the remaining ~98-99% is iron (not including contaminants). Say the outer two layers are both iron and have no carbon. When the layers are forged together, they are heated up to a high temperature and as Luftmensch mentioned the layers start to melt together. Carbon migration is when the carbon from the central carbon steel layer starts to spread out into the carbon layers. This occurs as the layers melt together.
You already have a sense for why this happens. Say you've got water and you drop a spoon full of salt into the water. If you quickly go in and scoop out some water from the same area you dropped the salt into, it'll be really salty water. If you wait some time, the salt spreads out across the entire volume of water. Sampling the water a few hours later, you'd notice it tastes much less salty. It's the same concept (diffusion).
...is it a good or bad thing?
This one is tough for me to say, since it's a very complicated effect. As carbon migrates, the effective composition of the steel changes. The organization of the atoms change. It could be good or bad depending on the situation...
How does it manifest itself in the forged blade?
Luftmensch gave a great answer to this in terms of aesthetics. I defer to blade makers to how it affects the properties.
Are some steels more prone to this phenomenon?
The closer in carbon concentration the different layers, the less it happens. The greater the difference in carbon concentrations, the more it happens. There's also a time dynamic.. if you haven't allowed enough time for the physical interfaces between the steels to melt together then it won't really happen at all.