I suspect that your problem involves bins containing different items. I haven't studied it closely, but here's how I believe it works:
1. A bin can only be accepted into a stockpile if each item inside it can be accepted. So if you tighten a stockpile's requirements you can accidentally kick an entire bin out of the stockpile because one item in the bin doesn't qualify. And you won't realize that the bin has been kicked out because it's still sitting on a stockpile square. But the game considers that bin to be loose change that just happens to be sitting on a foreign stockpile, waiting to find a proper home.
2. If a bin isn't accepted into a stockpile, then you can't do much with the contents. You definitely can't move the contents into another stockpile. I'm not sure if dwarves can use the contents, such as to put on armor or wield a weapon, but it wouldn't surprise me if they can't. Items in a homeless bin are pretty thoroughly inaccessible.
3. So you have two possible solutions to your armor problem:
a) The Usual Way. Put a stockpile under your mixed bin with settings broad enough to encompass everything in the bin. Then set up a second narrower stockpile to take the item that you want. Then set your "take".
b) A Clever Way. Have your soldiers strip, forbid all your other armor, and then tell one of your soldiers to get dressed. See if he'll pull the armor out of the homeless bin. Then have him strip again. Now the armor should be free of the bin, and your dwarves should move it to your target stockpile. If you try this, tell me if it worked. I'm curious.