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I am in the process of installing base cabinets in my kitchen. From what I have read online from articles such as this, it sounds as if base cabinets only need to be fastened to each other and the wall, not the floor.

This sounds reasonable, except that my kitchen floor has a considerable amount of runout (1-1/4" from highest point to lowest point, on opposite ends of a 11'x12' room). In this scenario, the cabinets in the low spots would be floating if only secured to the wall. It is my plan to make some custom blocking/shims to bring these cabinets level and to the proper height, but it doesn't quite seem sufficient just to have these blocks floating between the floor and the cabinet.

Is there a standard practice for this scenario? My thought was to at a minimum secure the blocks/shims to the floor with fasteners or construction adhesive. However, cabinet to floor I don't have many options at all due to the construction of the cabinet. It's made from 1/2" plywood and the floor of the cabinet is ~6" from the actual ground it rests on, so I would have to make some sort of blocking that goes behind the sidewalls to screw in from the side or blocking that reached the bottom of the cabinet to secure downwards from the cabinet floor.

Also, for other cabinets were only small, wedge type shims are going to be used, do you just break them off in place and leave as-is? Seems like over time these could walk out with only a thin 1/2" plywood sheet resting on top of them.

Edit: After looking at some more tutorials, it seems this one suggests that you run screws into the toe-kick at a downward angle to sort of toe nail it in place. Seems a little tough to do through 1/2" plywood and not split it, but if it's only to hold the shims in place that bear the weight of the cabinet that make sense. I'll just need to use longer screws on the cabinets that are shimmed more.

Shims are ordinarily left to just friction for anchoring. In your case, even if you only glue them to the subfloor they're unlikely to move. You could certainly also glue them to the cabinet base.

I wouldn't bother trying to screw them in, though. That's overkill alongside construction adhesive and will probably result in a bunch of split wood and swearing. You might pop a few trim nails into them if your have an air gun on hand, just to secure them while the adhesive sets.

In my current house, the cabinets are supported on legs. These have limited height adjustability, and then the toe-kick is non structural and clips to the legs.

Where the cabinet is supported on side-walls (or even a structural toe-kick) that come down to the floor, then one needs to insert shims under the walls in order to keep the cabinet level. In this case there is no need to worry about the shims wandering - the weight of the cabinet will hold them securely in place.

I'm not sure how well versed you are with this type of construction but have you considered framing out the portion of floor that will be under the cabinets and pouring some self leveling concrete??? 1 1/4" is a lot of shimming. This might save you a lot of time and give you great results.... just a thought.

I did an L shaped kitchen that had a drop in the pit of the L of 2.5". I made my own kick and shimmed it level. I didn't screw the kick to the floor but I did screw the base cabinets to the kick. I also don't typically screw base cabinets to walls but I suppose if you were to do a straight bank of cabinets and pulled all the drawers out with heavy items it is possible the whole bank of cabinets might tilt forward depending on how heavy the counter top and anything on the counter top was... being screwed into the kick unless the top drawers were super weighted. I typically toss ikea legs/rails/etc.

Screwing through 1/2" plywood is fine. Most of the time cabinets are made from mdf and even that accepts screws with no problem. It isn't like you are screwing in line with the layers of the ply.