Try to work within a small section of the game at a time, it will then take less moves to lower the Pets, and moves are very limited in this game.
Only one Pet can drop into a column at a time, but you need to get more than that down in a move if you can.
To do this, start above a vine locked virus block, then get the Pet to shift to the left by making space in that column (a rocket can help to do this).
Try to get a few Pets this way.
Attempt to have at least three, preferably four Pets in the one column.
Then also get some coming down in adjacent column/s.
Rescue these, then do again and try to as many as you can of the remaining Pets on the board before dropping them.