I'd suppose this depends on what you view as moral behavior but I thought I'd ask the question to see what people would say.
It indeed depends on what ones standards are and what parameters would be acceptable.
Also, and I don't think this is an unimportant factor, one would also have to consider what the motivation of the actions are which are going to inform the moral judgement.
For example, if a certain immoral action wasn't motivated (or obstructed) by either theism or atheism... then did one of them "lead" to that behavior?
Your question, after all, asks about direct relationships. Causal relationships between (non)beliefs and behavior. So I don't think it would be fair in that case to take mere correlations and then say "X leads to Y".
It is very easy to say that "atheism leads to immorality". But it's kind of hard, imo, to say how mere disbelief of rather specific fantastical claims in any way form the basis or motivation of certain specific seemingly unrelated actions. I've never seen someone rob a bank "in the name of disbelief of X" for example.
Is the world becoming more moral or less moral?
I'ld say that it becomes more moral overall.
There are bumps on the road and localized descends into immorality. But overall, in the sum of all things, I'ld say the trend is towards more moral.
And, does this have anything to do with the decline of religious belief?
Indirectly perhaps, but yes, I think so.
But I don't necessarily view this as a causation (hence "indirect").
I think it's more a correlation then anything else.
It's about people becoming more aware of reality, the world around us, the true nature of what it is to be human. And as it happens, such information usually flies in the face of (at least literalist) interpretations of certain mythological religions.
So the trend towards more morality isn't imo caused by a decline in religious belief.
If anything, I'ld expect it to be the other way round: the decline of belief being a result of the same cause of the rise of morality. One doesn't trigger the other. Instead, both are triggered by the same underlying thing, which is overall progress in science, general knowledge and a better realization of our place in the world and the universe.