Best way to create non-character animations?

It seems like most of the tutorials I’ve seen involving animating in Unity involve character animations and the like. I would like to create a game that has sort of “set” animations that have less to do with game pieces and characters and are just “there”.

To get an idea of what I’m trying to do, here’s a video of gameplay from Disney Emoji Blitz.

What I’d like to do is have animations such as at 1:28 and 9:01, where little images pop up in front of the board and are not involved with the game pieces themselves. I’d like to be able to rotate, transform, and resize/“squash” the animations (such as with the smoke pipe at 9:01- notice how it sort of stretches upwards).

Does Unity natively support this kind of animation, or would this better be done in an external program like Flash?

Please let me know if I need to further clarify.

Yes - you can create these animations in Unity or external. For more precise control over the animated elements you might need a bone hierarchy or deformers which are not in vanilla Unity, but there are a ton of assets in the store that can provide even more enhancements like bone hierarchies, deformers, tweens and reactions to physics, adding even more complexity to animated elements than you can do in other packages.