My approach has similar aspects than many have mentioned.
Almost always, it goes through paper. I have a stack of index cards everywhere: on my night desk, at my desk in my home office, at my desk at work office (when I have one), in my wallet, in the back pocket of my notebooks, and so on.
When a random idea pops up, I jot it down. At home, I have a small box at my desk where I throw all my fleeting ideas that I have written on these notes. Then regularly I go through them and decide whether they are worth keeping and transferred to my digital system or thrown away if the idea doesn’t resonate anymore or it got solved before I got to it.
Personally, it’s a key thing to do this on paper: if I open my phone or my laptop, I’m so likely to get distracted by a thousand things either before or after I make the note. When it’s just a piece of paper, it’s written and then I’m back to what I was doing.
If I’m with other people, these days I think people feel less bad if I take out a paper, write on it and put it away in the middle of a dinner than if I’d take out my phone and start browsing. I don’t know if people care like that but it’s one reason why I prefer paper notes in general when I’m with others.
Whatever method I use, the most important bit is regular review. If I let those fleeting ideas pile up in my inbox at my desk, they become a chore or I get overwhelmed. I maintain similar habit/workflow with my read-it-later app where I save articles/websites as I read and browse the web and with Mastodon bookmarks. I keep the lists clean by going through them regularly so they end up in the right place and don’t pile up.
This review also lowers the barrier of starting to make a note. When I make a note that’s temporary, there’s no judgement of whether it’s good or important enough. Sometimes it’s very temporary like a braindump you mentioned and sometimes it’s a start of something that grows into an article or a zine. I delay that judgement to my review to keep it easy to get started with writing down ideas.
Here’s an example from today: I was watching a Youtube video on Magic the Gathering and something they said triggered an idea. I had written about something tangent to it earlier and wanted to add that new thought into that previous note so I took out one of my index cards, wrote down what the idea I got was, where I got it from and what I wanted to do with it. Then I put that back into my notebook and continued watching the video itself. After that, I could focus on the rest because I didn’t have to think about the branching thought because I knew it was saved to be fleshed out later.