I wouldn't hundred percent subscribe to Tamil (refering here to classical Tamil Sangam poetry) being mora-timed. The counting of morae (from the first grammar Tolkāppiyam onwards) is somewhat confusing. E.g. there is of course one mora allotted to a short syllable and two morae to a long one, but moreover there is half a mora for overshort u and i and even a quarter mora to an overshort aytam and so on. The terminology of morae seems to be superfluous in Tamil and probably taken from Sanskrit, as can be gleaned from the borrowing of māttirai from Skt. mātrā - mora.
Basically Tamil meter is based not on a dichotomy of long and short syllables but on one of simple and compound metrical units, called nēr and nirai respectively.
Compound unit: one short syllable followed by anything.
Simple unit: the residue single syllables.
These units are then arranged in bars, for example bars of two units each and e.g. four bars to a line. It therefore does bear some likeness to Middle High German meter, which is based on bars and it indeed does insinuate moric equivalence of one long syllable to two shorts.
Now, syllabic meter can be considered native to Indo-Aryan, since already Vedic Meter is based on the distinction of longs and shorts, though with much less strict patterns than Classical Sanskrit meter. It seems rather likely, that the moric meters of early Middle Indian and then Classical Sanskrit evolved by midwifery of Dravidian meter.
Cf. for Tamil Meter Kamil K. Zvelebil - Classical Tamil Prosody, New Era Publications, Madras 1989;
for suggestion of Tamil meter being purely moric and bar-timed George L. Hart - The poems of Ancient Tamil, University of California Press, Berkeley 1975.
For an overall discussion of the evolution of Classical Sanskrit Meter there is an unpublished master thesis of University of Tübingen in Germany, which I could provide, if German is an option.