Medium boards use the easy moves too, but somewhere they run dry and you have to read a whole row or column at once.
Take a row that reads cat, four empty squares, cat. It needs one more cat and three dogs. If the cat goes in the second square, the three dogs land side by side in squares three to five, which breaks the rule. The same happens if it goes in the fifth. So the cat sits third or fourth, and either way squares two and five are dogs. You still don't know where the cat goes, but you have placed two dogs.
That is the medium move: run through the ways a line can still be finished and fill in whatever they all agree on. With three or four empty squares there are only a handful of ways, so it is quicker than it sounds.
When the easy moves stop, look for the row or column with the fewest empty squares and count its options. One placed animal is usually enough to start the easy moves again.