Sometimes I notice that I am about to fall asleep, but I don't often remember the actual falling asleep part. Being about to fall asleep mostly feels like getting more and more relaxed, and my train of thought slowly getting derailed. (I can't remember what I was thinking about from the beginning to the end of a sentence, and I get randomly selected nouns and verbs, some of which are just stolen from what ever background podcast/radio noise there is.)

Also, I usually remember the 'about to fall asleep' stuff better when it happens in the mornings after I've woken up and decided to go back to sleep, then when it happens in the evenings when I first go to bed.