I'm only aware of the old papers by Champion on African Melyris s.l. and the recent work of Patrizia M. Estrada and Robert Constantin on Melyridae s.str. genera of South America.
@Vladyslav: What do you mean by Asia? Do you include Turkey & Iran or India?
I have papers by R.Constantin on Melyridae from S.America. I need papers for Central Asia, Middle East, Arabia, Iran, Pakistan, Afghanistan. But, as I see, is not so many publications for this territories, unfortunately.
MAJER K. 1990: Results of the Czechoslovak-Iranian entomological expeditions to Iran 1970, 1973 and 1977. Together with results of collections made in Anatolia. Coleoptera, Melyridae: part II (Melyrinae, Cerallini). Acta Entomologica Musei Nationalis Pragae, 43: 163-181.