Navajo would be one to look at. It is still clearly an agglutinative language, but it's well along the pathway that's hypothesized to lead to inflecting morphology.
The Wikipedia article — which FWIW is pretty decent — puts it this way: "Navajo is an agglutinating, polysynthetic head-marking language, but many of its affixes combine into contractions more like fusional languages."
To pick out an example from Wikipedia: the underlying morphemes di-'a-ni-sh-ł-bąąs, after various phonological rules have been applied, give you the word di'nisbąąs. Just looking at the surface form, you can't segment it cleanly into six morphemes. (There's no clean segment of the surface form corresponding to the underlying -ł-, for instance.) At best you can split it into three "chunks," some of which correspond to several morphemes and contribute several grammatical features to the word.
Eventually, the hypothesis goes, speakers will cease thinking of the word as having six morphemes, and start thinking of those three "chunks" as morphemes in their own right. Keep this up over a few more centuries and you'll have a truly inflecting language, where each word is made up of just one or two "chunks" and each of the "chunks" has a complex meaning like "third person singular past tense."
