By this point, they are very old, corroded and don't put out a lot of light. It is very difficult to change the light bulbs. So the thinking was to replace them.
We bought a new fixture from Home Depot -- Milford 4-Light Brushed Nickel Flush Mount -- manufactured by Livex Lighting. This is an open design with four candelabra (E12) base lights.
Circuit breaker 30 controls these lights, so after turning that off, we removed the old fixture and installed the new one.This took about 30 minutes.
We put 4 "60 Watt" LED bulbs into the light fixture. Each of these takes about 4.5 watts, so that's only 18 watts total, well under the 40 watt maximum. But they put out a lot of light. In our case we got "Daylight" bulbs (5000 degrees Kelvin) putting out 500 Lumens. With 4 bulbs, we get 2000 Lumens, which is very bright.
Update: This worked well, so we bought another one of the same light fixture and installed it on the other end of the back porch.
For this one, we put in one Daylight bulb (since we still had one left over) and then put 3 soft-white lights (3000 degrees Kelvin) in the others.