One of the most marked differences between Old and Modern Kannada, is the widespread (but incomplete) phonological change from ಪ್ p to ಹ್ h. This'll be a short thread on a theory by Edwin Tuttle (American Journal of Philology, 1929) that seeks to explain this phenomenon.
Tuttle says that this shift was the result of a preceding phonological change, from ವ್ v to ಬ್ b - the 'pre-Old Kannada' of the earliest inscriptions has e.g. ವನ್ದು vandu (having come) ವೆಳೆಗೆ veḷege (may it grow) etc, which soon become ಬಂದು bandu ಬೆಳೆಗೆ beḷege in Old Kannada.
The Kannada now having p-b minimal pairs (i.e. words that only differ in which of p or b they have), speech evolved to exaggerate and make the differences between these clearer, by aspiration of the former i.e. ಪ್ p became ಫ್ ph. The latter developed into ಪ್ f, and finally ಹ್ h.
Personally the ಪ್ f seems unnecessary here, ಫ್ ph → ಹ್ h makes sense if I try to speak these changes out. A parallel to this can be found in Indo-Aryan e.g. the bh (an aspirated bilabial stop like ph) in Sanskrit bhū (to be), evolved into h e.g. Hindi hōnā Marathi hōṇē etc.
Citing this parallel, K. V. Subbayya (Indian Antiquary Vol. XXXVIII, 1909) conjectures that this occurred in Old Kannada due to the influence of Marāṭhī (the term 'Prakrit' would be more appropriate here no?). I don't see why this couldn't have arisen in Kannada independently.
Anyway, this is the only theory I have come across that tries to explain this phenomenon. The major Dravidian linguists seem to have non-Kannada backgrounds/interests, thus its features are generally minimalistically dealt with in their work.
Lastly, I've never seen ಫ್ ph (and certainly not ಪ್ f, no native character for this exists in the script) as an intermediate stage in place of ಪ್ p in any inscription I've encountered to date. I don't think the epigraphical evidence supports this theory.
