First of all, a note: not everyone believes in clitics! Some theories don't include them as a separate concept at all. But for those that do:
The defining feature of a clitic, in most theories, is that it's tightly bound to the word next to it. Phonologically, it's treated as part of the same word, rather than a separate word in and of itself.
But, crucially, clitics can move around to different places in the sentence—they're not morphologically or syntactically part of the same word. Compare the following:
Alice and Bob 's house
Domus Aliciae et Robertī
In English, the 's attaches to the entire phrase "Alice and Bob"; in Latin, the genitive (possessive) marker is put onto every word in the phrase instead. So 's is called a clitic and -ae is called an affix.
As an additional guideline, clitics are "promiscuous": they can attach to anything and everything that's in the right syntactic position. Affixes generally aren't: the English plural -s can't attach to nouns like ox or man, for example, while 's can attach to the end of any English noun—or any phrase that acts like a noun, which is why we see surgeons general but surgeon general 's. (It's not always written as 's, but that's just orthography: Jesus' is pronounced just like Jesus + 's.)
So while I don't speak Danish, I wouldn't say that sort of movement disqualifies something from being a clitic: it's actually a point in favor of the clitic analysis, as opposed to analyzing it as an affix. Whether it's a clitic or a separate word then comes down to phonology.