All of the plants that you mention can become invasive. The blackberry and raspberry spread by canes, so you can usually keep them under control by digging and/or pruning out the unwanted shoots. Mint spreads by runners, so it needs to be kept confined in a container. Porcelain berry spreads not only by vining, buy by seedling berries. So until the main vine is removed, you'll have a constant source of new plants.
If they have taken over the area, it may be best to kill them all off with a nonselective herbicide such as roundup, then use the landscape fabric and mulch to keep them out of the beds.