Monster Baby Rescue!

From BoardGameGeek:

In Monster Baby Rescue!, players take care of their sad, lost and ill "Baby monsters" who got lost to us from fantastic world. Every player chooses their own "Monstie" - Dragon, Mandrigora, Ork, Basilisk, or Kerberos - and with the help of tiles displayed on the table takes care of it. There are tiles to groom particular parts of bodies, tiles with playgrounds, cosy places, diamonds to play or decorate with, and magicians (vets). These tiles have different "price" which is paid by different number of steps of your Monstie on the time track. The next player is the one who is the most behind (takes the cheapest tiles from the table). Players gain points for bonuses on tiles and also for the levels, they groom, improve, and take care of their Monsties. -description from the publisher