Do you think little children can get framed for a very serious crime (like maybe murder) so easily?

I think the presumption of innocence is especially sttong for individuals who are far more likely to be the victims of serious crime than the perpetrator. There would have to be a massive weight of evidence that points specifically to the child having committed the crime and noone else before any reasonable person would even consider such a thing possible

Absolutely. It happens all the time, someone framing a kid, especially a young kid, because the young kid is far less likely to be held criminally responsible and, if the they are, far more likely to get little to no incarceration and instead get rehabilitation instead of penalization. Also, framing a child, especially a small child, has the added advantage that they are highly ineffective in aiding in their defense because they so often fail to be able to articulate themselves in a coherent manner to adults, which is exacerbated by their inability to grasp what's happening to them or their grasping so little of what's happening to them.

The hard sell, of course, is getting a prosecutor, judge, and jury to believe that a child who didn't actually commit the crime, even if framed up well, is physically and mentally capable of having committed it, like, yes, you can put the handle of a bloody axe in a child's hand so their finger prints are in the blood on the hand and make sure they have no alibi and arrange it so there's some plausible story for the child doing it and no possibility that the prints got onto the handle in some innocent way, but in court, when the child can't even lift the axe off the table because it's too heavy and when the jury looks at the face of a genuinely clueless child who never killed anybody, the whole thing tends to fall apart. 

But, yeah, framing kids for serious crimes is a thing. Street gangs are particularly notorious for doing it. Sometimes framing works. Sometimes it doesn't.

By the way, another version of framing a child for a crime is making a child actually guilty of the crime by mentally and emotionally manipulating the child into committing the crime and believing they did it of their own volition. The naïveté of children make them particularly vulnerable to this. Fortunately, psychologists can normally suss that a child who can be so manipulated still lacks the mental capacity to know the difference between right and wrong and so lacks the mental capacity to form criminal intent, thus making them not criminally responsible, even without actually sussing that child has been manipulated or sussing who manipulated the child.

Anyone can be in the wrong place at the wrong time, and be falsely accused. But framed? Who would do that?