As so often with English, context is extremely important.
In many dialects of English, the last pair of examples are interchangeable - though in any given dialect, one may be more natural or common than the other. Which order is more common will depend on what 'that' is, as well, and what the verb is, what the adverb is, and what the demonstrative refers to. Remember that some combinations of verb and adverb amount to a distinct phrasal verb, as well.
That will be true for a lot of choices of verb/phrasal verb, but not all of them. Your first example is one of the exceptions. The second would suggest that Kate had delivered something, a package or whatever. The second could mean that, but could also mean that Kate fell from whatever this refers to. You could usually tell the meaning from context, but there is ambiguity there. Perhaps for that reason, the second is more generally used in my experience (British English).