We have been very critical of the OIT position, but just to be fair let us consider their position. This is only fair, since we have been treated to Renfrew’s views on an Anatolian source. In general the sources of PIE and proto-PIE have to be somewhere, so….
The quote from Elst shows a clearly falsifiable set of hypotheses. So we should let the debate play itself out. Note the similarity and equal difficulty of both claims. I think the OIT have failed to consider language change rates and wish their Vedas were in Sanskrit in the fourth millennium BCE. That won’t work.
“The emerging alternative”
Koenraad Elst summarises “the emerging alternative to the Aryan Invasion Theory” as follows. 
During the 6th millennium BC Proto-Indo-Europeans lived in the Punjab region of northern India. As the result of demographic expansion, they spread into Bactria as the Kambojas. The Paradas moved further and inhabited the Caspian coast and much of central Asia while the Cinas moved northwards and inhabited the Tarim Basin in northwestern China, forming the Tocharian group of I-E speakers. These groups were Proto-Anatolian and inhabited that region by 2000 BC. These people took the oldest form of the Proto Indo-European (PIE) language with them and, while interacting with people of the Anatolian and Balkan region, transformed it into a separate dialect. While inhabiting central Asia they discovered the uses of the horse, which they later sent back to the Urheimat. Later on during their history, they went on to occupy western Europe and thus spread the Indo-European languages to that region.
During the 4th millennium BC, civilisation in India started evolving into what became the urban Indus Valley Civilization. During this time, the PIE languages evolved to Proto-Indo-Iranian. Some time during this period, the Indo-Iranians began to separate as the result of internal rivalry and conflict, with the Iranians expanding westwards towards Mesopotamia and Persia, these possibly were the Pahlavas. They also expanded into parts of central Asia. By the end of this migration, India was left with the Proto-Indo-Aryans. At the end of the Mature Harappan period, the Sarasvati river began drying up and the remainder of the Indo-Aryans split into separate groups. Some travelled westwards and established themselves as rulers of the Hurrian Mitanni kingdom by around 1500 BC (see Indo-Aryan superstrate in Mitanni). Others travelled eastwards and inhabited the Gangetic basin while others travelled southwards and interacted with the Dravidian people.