First, fish can become acclimated to humans in a way that resembles the beginning of domestication.
My partner keeps reef aquariums, and the fish in those become conditioned to approach for feeding and scratches with a small carbon fiber rod (this probably resembles the activity of cleaner shrimp, at least to a fish brain). I believe it's likely that any animal smart enough to hunt for food (which includes most reef fish and nearly all cold water ones) is smart enough to accept this level of conditioning, and after generations of selective breeding (the ones that become conditioned get to breed more successfully, because they and their fry get fed more) they will become domesticated.
The bad news here is that there are few if any cold-blooded species that transition from warm to cold water in nature. Tropical fish, those that live in water around 20-24 C, will quickly sicken and die if the water temperature gets too low, while cold water fish (salmon, pollock, etc.) cannot survive in water that's too warm. Tuna might transition -- I don't know their ranging habits well -- but they'd be an exception (they are anyway, because they aren't fully cold blooded). Herring (cold water schooling food fish) and sardines (warm water schooling food fish) are similar, but as far as I know, neither migrates from one climate to the other.
So, yes, it's very plausible to domesticate fish over a period of many (fish) generations -- but whether the herds could be moved from cold to warm or warm to cold is highly questionable.
Now, co-domestication might solve part of the problem -- seals in cold water, or sea lions in warmer water, could be domesticated (over even longer time frames), possibly even to the point of maintaining (or at least confining) a fish stock in one climate while the merfolk are in the other. These sea mammals are comparable in intelligence to dogs, are (or were) often seen in circuses, trained like dogs and horses are trained -- but not so intelligent (like dolphins or whales) that questions of enslavement arise.
Alternatively, there might be a need for a small number of merfolk to remain behind in one climate while the bulk of the population is in the other, in order to maintain the herds (or supervise the seals and sea lions). This might become a "coming of age" rite of passage -- staying behind is how you become an adult.