"Desi" is a term popular amongst Pan-South Asians. It is used to refer to Dravidian, Dardic, and Indo-Aryan speakers. But strangely enough it does not apply to Iranic speakers (ie. Balochis, Pakhtuns) despite Iranic speakers in Pakistan sharing common linguistics, genetics with Indo-Aryan and Dardic speakers. (Search Indo-Iranic languages).
The word "desi" has no scientific acceptance in modern-day anthropology or linguistics. A Dardic-speaking Kashmiri has no linguistic relation to a Dravidian speaking Tamil. Dravidian languages belong to a completely different and un-related language family than Dardic and Indo-Aryan languages. Dardic and Indo-Aryan along with Iranic are part of the Indo-Iranic family of languages.
What's more is that genetically the Dravidians lack R1A genetic markers that are least found in Southern India (though some sources state Tamils have a significant R1A contribution than other Dravidian speakers; suggesting genetic contributors in their gene pool coming from more northwards) while Dardic and Iranic-speakers in Pakistan have it the most.
So clearly "Desis" are no more than a people of an imagination based on ignorance, pseudoscience and false political propaganda.