This is an interesting question. But seeing as you're a family member of an Aspie I could see why you would like some description. Which I don't mind giving, or answering any further questions; mine can get pretty intense.
So, I consider my more milder "meltdowns" anxiety attacks. Whether it's due to a social situation or high stress levels, I will usually feel very very warm, especially in my face and ears. I feel like I can't breathe, and want to hide somewhere safe. Sometimes I even have stomach problems.
Now, my more intense ones I have experienced since I was little. I get super upset, often over really small stuff, and even if it may be triggered by someone else I always end up internalizing it somehow. I cry of course, and for some reason this is really hard for a lot of people to understand or handle. I can't help crying though, and it literally hurts if I try not to. When I was little I would proceed to pull my hair or bite my tongue. As an adult, I will often feel sick and want to throw up, but since I can't I will make myself throw up. Since I naturally have long nails it's really easy to scratch myself, usually my face. Sometimes, I will hit myself.
This stuff may sound pathetic, but I honestly feel like I can't control it usually. For instance, one time I remember very clearly, I was arguing with my mother (probably about something silly), and I said something that really hurt her feelings. I immediately felt very bad and started screaming for her to forgive me, crying, and hitting myself. It was almost as though I was outside of myself looking at the situation though, and I couldn't control what was going on even though I wanted to.
Thankfully I haven't had one of the intense meltdowns in months. It is very exhausting when I do though.