There are so many ways I could answer this, but right now, this comes to mind:
When we are met with stressful situations (caused by our thoughts ... always that way), it tends to be because of some need to be loved or approved of or linked with appreciation seeking. This tends to be unconscious and quite automatic that we even hardly notice we are doing it.
From a young age, there are certain thoughts that have been programmed into us, such as:
"I need to be nice so that people like me"
"I have to be polite so people think I'm a good person"
"It's important to be loving/kind/generous" etc.
"People love considerate people"
With those sorts of thoughts (and there will be hundreds similar to those) we end up having this built in part of our personality that's designed to please others, constantly watching for signs that we're succeeding.
This can end up with us behaving in a totally fake way .. not saying what we really mean, not acting out of honesty, not living in our own peace ... instead, we focus and care more on how the others respond to us, whether they approve or disapprove. This outward focus is stressful, because in some way, we transform ourselves, acting out that it's OK, being polite and tactful, trying to control and manipulate our impression on others, not acting with integrity. So, instead of living out our life, we just act it out.
When I'm confronted with others mood swings, and I don't want part of it, I just walk away. I don't make polite excuses even ... I might say: "Oh I see you're having a bad day, so I'm off, just so you don't spoil mine!" ..
What they think of me is none of my business, but my feeling good in myself, my peace and health is MY business.
So, from my experience, being vulnerable to other people's moods is linked with caring more about what others think of you (seeking love, approval, appreciation) than you think about yourself. Being Invincible in such a scenario is to turn the situation around, knowing that there is only one person that counts when it comes to loving, appreciating and approving of you, and that's YOU :-)
The book: "Loving What Is" By Byron Katie helped me a lot with working on a whole host of issues, I think you'd get a lot out of it.