Since some time has passed without an answer, I thought I'd try with a partial one.
I don't know of any examples of ablaut reduplications in English where the first vowel is the backer/opener one.
You asked about a "process", but I think it's difficult to analyze because reduplication is not extremely productive in English. The words that show this pattern of vowel change have somewhat miscellaneous origins, which makes it difficult to describe them in terms of derivation from some single process.
Some OED entries suggest that in at least some cases, the back vowel is original and the front vowel created during the derivation: the entry for (k)nick-(k)nack says "Reduplication of knack n.2, with first element lightened as in crick-crack, etc."
In some cases, however, each part of the reduplication consists of a preexisting word-form: this seems to be the case with sing-song and kitty-cat.
There are alternative types of reduplication; as mentioned in the comments, there is rhyming reduplication with consonant alternation. "Schm-" reduplication is, unusually, somewhat productive; there are also fairly non-productive but notable patterns like h-C reduplication and C-w reduplication (I wrote an answer going into more detail about these here: Why is a rhyming word beginning with “h” put before another word to create a new term?)