1. If the cost of a tile decreases, the precalculated heuristic will be too high, so A* might find a non-shortest but ok path. In game, you can think of the dorf as following the path they already know about, because they don't yet know that there's a shorter way.
2. If the cost of a tile increases, the precalculated heuristic will be too low, so A* will find the optimal path but it will take a little bit longer (still not as long as if we weren't using this heuristic). In game, you can think of the dorf as following the path they already know about, but running into a wall, so then they find a path around it.
But many games recalculate distance to target (one ping only) over and over again each step, so moving a single block half a map away causes an entire army to repath immediately.