Experts and BHG readers answer.
vine-like weed that is taking over flower beds
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.