The clip you provided is pretty low res to see clearly.
For the one at 0:09-0:14: It looks like they tried to combine two moving shots. Also looks like they had some trouble with a smooth transition in the motion so you can see where the shots change over. The whole clip has a blur effect on it while the 3d papers blow in the foreground. The blur helps blend the effect.
The second one at 0:24 looks like they just tried to line up the shots and faded between them.
You need to match your shot, lens, wardrobe and pose of everything you want to look like it's not changing while you fade over. However if you're shooting green screen, like they did, it will make it easier to fade the background layer while your foreground remains a whole clip.
There are varying degrees of complexity that you can reach with this effect ranging from easy to very difficult. Like most things, plan each transition carefully and you'll save considerable time in post.
Like Mike said it's a combination of production and post-production. For something like this you need a lot of planning and you need to know exactly what kind of gear you can get. If you can get a motion controlled rig for the camera that can match the move you do in the chroma studio shoot that you can shoot any background plate. If not, you'll need to shoot the actors in the studio first, track the camera move and then build a virtual background using images, footage or 3d apps for that.
You could shoot the footage in a high res, and then scale it down. This way, you can create the push-in by scaling up. Then you just use a freeze frame for the first bit and photoshop or roto out the background. Notice there isn't much motion on the subject until it changes? You can get away with that by adding particles and even using the puppet tool to move the subject ever so slightly. Then when you transition to the original piece of footage use whatever motion you want. Be aware of shadows as you remove the background though, you'll want to make sure your lighting is consistent if you really want to sell the effect.