How do I remove moss from an outdoor brick walkway?
My immediate vote is for a pressure washer. They are relatively cheap these days. Lacking that, a simple high pressure hose nozzle and a good stiff brush will do well too. Various oxidizing bleaches will also work, all of which will last for only a limited time.

To stop it from returning is difficult, since any moist, shady spot under a tree or bush is likely to grow moss and algae of this sort.

One idea is to remove the overhanging growth from the planters. Trim them back. This will help the brick pavers to dry out, and will reduce the tendency for unwanted growth.

Another idea is to get creative, and buy some zinc strips. Copper flashing might work too, but they sell zinc strips specifically for roofing applications, where moss and algae tends to grow on roofs. The elemental zinc leaches out (very slowly) when it rains, which in turn inhibits growth of the algae where it leaches out. So you could afix these strips to the bottom edges of your planter. Make it decorative and it should help, although I cannot assure it will work perfectly. At least this will inhibit the moss.

Another idea is to buy/make a few copper pots or planters. Place them decoratively right on top of the problem spots. As the copper leaches out from weathering over the years, it too will inhibit growth on the walkway, you not be a problem for larger plants in the area. If you cannot find a pot you like, you can make a planter from wood, but then wrap copper flashing around it, nailing it in place.

Be creative if you wish to inhibit the growth.