Why would you have gaps? With LED bar lights, you will wan to fill in as closely as possible. Most of them can be coupled together and then linked to each other to get too the next space if your caninets have stiles under them, they can be notched. Many of the new ultra slim LED strips can even be fitted in the notched stiles. I would put the trim on the bottom of the cabinet unless you have sleek ultra modern cabinets. Then use the recessed bottom. Remember to put your light stips at the front of the cabinet and not the back. Just space them as evenly as possible all the way around filling in as much space as possible.

Hafele has several aluminum channels that are made to recess. Some come in specific lengths with lights already in them, several are designed for use with the tape. For 3/4" thick bottoms there is one that lays flat and protruded 1/16", another tilts slightly toward the backsplash so it sits proud 1/4" at the front. All come with diffusers. They also have several pucks that recess. The hard part with recessed is locating the wires. AS LWO suggested there are a number of nice looking ones that can be exposed. I just finished a white high gloss slab project where we had the doors set lower than the box, ordered solid stock with a bevel on the back, set the solid stock behind the doors and mounted the light rail to it. Advantage is easy and simple installation, clean look both open and closed, cleaner and cheaper than recessing the bottoms of the cabinets. Downside- you give up 3/4 to 1" in height inside the cabinet and of course you have to be working with a company that does that.

Thank you so much. I may need to shop for another electrician. I can understand why one would not want to "finish" work that is hidden. I don't know why this time the electrician is being so difficult (like he's asserting the right to control everything). All I can think of is that something unfortunate is going on with him, but that of course is none of my business. I pay immediately and without quibbling, in case you wondered, so it's not that. I really dislike working with electricity and the few little things I do like change out a receptacle or switch, I shut off the whole house at the breaker box. As far as surface mount, I know it's ugly but the cabinets have cabinets abutting on both sides so there can't be an inset J box. A wall switch would work but I did not yet have the wall (really a short extension of an existing wall) framed out and he was pressuring me really hard to decide right away to either put the switch on the existing wall (which would have been quite an awkward location) or inside the cabinets. I appreciate your taking the time to answer me. It has helped me reach the conclusion that I wanted to avoid, which is trying to find another electrician.

No need to upload pictures of all 5 because they are all situationally identical to the picture attached here. The cabinets have not been installed yet. In this picture you see the underside of the existing upper cabinet and the space where the old lower cabinet once was. You also see the outlet which is approx. 7" below the bottom of the cabinet. All of the j-boxes will need to be replaced as they are from the early 80s and suffer from plastic fatigue and will no longer hold a standard wiring device mounting screw. Replacing them with a 2-gang is not a problem. The question is not how to connect everything but rather how to get 5v to the underside of the cabinets. There is no space above the cabinets, they go to the ceiling. This wall is an interior wall as are 2 others. Two of the outlets are on exterior walls. Logistically, putting a central source of low-voltage power and distributing it to the underside of the cabinets through the walls is not possible. I floated the idea of mounting an outlet inside the cabinet and using a wall-wart to supply power was poorly received by the client. I even suggested a recessed A/V j-box where the power supply would not encroach on the interior of the cabinet (such as an Arlington DVFR1W-1). That was only slightly less poorly received. If there is a code-compliant way to put a 5V (or even 12 V) class-2 power supply into the LV half of a dual j-box, I'd like to hear about it.