The creature either falls or clings to your new form
Per the PHB, p. 67:
You choose whether your equipment falls to the
ground in your space, merges into your new form, or
is worn by it.
No mention is made about creatures. It wouldn't make any sense if they were absorded into your body. It's definetely safe to assume they aren't directly affected by your shapechanging at all.
If your hawk pet is standing on your shoulder, it either stays firm on your "new" one, or it is forced into flying (or fall in a... Hawkward manner), depending on what beast you transform into. If you are carrying a puppy in your backpack, the backpack is absorbed into your new form and the puppy most likely falls on the ground. You can't suddenly transform while shaking hands with someone and having their hand fused with yours. You can burn it if you transform into a Fire Elemental, though. The creature suffers the indirect effect of being in contact with your body while it changes shape, but it doesn't get involved in your transformation.