Dialectical materialism can be applied to basically anything and it takes awhile to develop it. I'm still not an expert by any means but I have certainly built a better understanding of it over the last 6 months as I've touched grass a lot, lot more and also re-read a lot.
It's a pretty broad question, asking how we apply it to broader society. So I'll just do an example I guess.
When looking at an issue, the first thing(s) I look for are the contradictions within it and the contradictory relationships it has with other things. We know literally everything is interconnected, and that everything exists in a state of contradiction until it is resolved (and then transforms into a different contradiction but blahbababa) so that's where I usually start.
Thinking about a gas station near my home, in isolation, is just a gas station. It operates perfectly well and statically to our liberal friends or if you're not paying attention. But taking a diamat lens to it I would say that there exist tons of contradictions and interconnected relationships that drive the success or failure and the overall conditions of that station. There exists obviously the contradiction between the workers and capital owners there, the one we all know. This contradiction, the primary one in economic life under capitalism, sets the stage for others to pop up - a worker may recognize the futility in how hard they work there, and leave certain tasks undone. Contradictions like that within the gas station build up until they are resolved by either another worker taking the task up, the worker getting fired, or people neglecting to come to that station if it's bad enough, leading to its closure. This is not static, and is constantly developing and in relation with everything else in the material world.
Connecting the gas station to society is easy from there, you just take off the magnifying glass and look outwards. The station exists in one part of a country that operates under that same economic system. So more contradictions pop up that are not internal to the station. Perhaps the piping that flows the fuel is not properly maintained because those workers are on strike, and ceases to function. The station is now defunct due to contradictions of the economic system, due to no fault of its own other than existing under this material reality.
Everything that happens affects everything that happens. The universe is a dynamic interconnected place and using the diamat framework is immensely helpful for making sense of things.
I could give a kind of abstract math equation, which can be helpful, but for me examples usually do the trick better. The more examples you hear the easier it will be to formulate your own and yeah.
Nobody has dialectical materialism "figured out" either, btw. The best analyses require many people, many years, much PRACTICE -> THEORY -> PRACTICE etc.
Hope this made sense, and if anyone with a better understanding of this than me wants to chime in feel free.
Yayyy everyone got a second job :D this must mean economic prosperity for the people