Hey! I work with Premiere Pro so maybe I can help you out.
Hmm... Since premiere pro doesn't really have a 'cross blur' transition, we'll have to make it ourselves!
Using the razor tool, razor the last 15-30 frames from the end of the first clip and razor 15-30 frames from the beginning of the second clip. Now let's apply a blur. Using keyframes, we can get that blur transition you want.
Start on the first short piece. Start with zero blur, and end with 100 or so. Then go to the next piece. Start with 100 blur then go to 0. Done!
If that's too confusing I'd be more than willing to send you a video showing how to do it.