I've come across sources that both either dismiss the theory as mostly fiction or claim that martial arts and military strategy was passed from ancient India to the East.
It's essentially all fiction - "mostly fiction" is too kind.
The classic version of the specifically kung fu story can be summarised as:
Once upon a time, there was kalaripayattu, the Indian martial art from which all others sprang. This, or one of its children, was brought to China by Bodhidharma, the great Indian missionary who brought Chan (Zen) Buddhism to China. Disgusted by the poor physical fitness of the monks at Shaolin, he taught them exercises derived from martial arts, and hence Shaolin became the birthplace of kung fu (and the centre of Chan Buddhism in China).
Some versions continue, and link this with later schools of martial arts:
In such-and-such dynasty, government soldiers came and destroyed Shaolin monstery, and the monks dispersed. One monk, <insert name here>, went to such-and-such province, and became the founder of such-and-such kung fu style.
For the first part - the Bodhidharma/Shaolin part - the martial arts component of the story is fiction. Shaolin gained its fame as a martial arts centre quite late (during the Ming), and the link between Shaolin martial arts and Bodhidharma only appeared in the 16th century. This was in the form of the attribution of the authorship of the Tendon Change Classic (AKA Sinew Transformation Classic) to Bodhidharma. Shaolin had earlier connections with fighting and warfare, but this appears to have been unrelated to (unarmed) martial arts, simply being a case of Shaolin using its private army (as discussed in https://www.reddit.com/r/AskHistorians/comments/8nk3p2/why_did_orders_of_monks_and_monasteries_in_china/dzwm0t7/ monasteries were often major landowners, and would form private armies to protect their holdings, just as other major landowners did).
For the second part, once Shaolin was famous as a martial arts centre, various martial arts styles/schools would claim to have originated from Shaolin. An exception to this were the "Daoist" internal martial arts, claiming to have been developed by legendary Daoist immortals (despite having, as at least one contemporary writer noted, a body of technique that was essentially the same as the Shaolin-derived styles).
Regarding more general claims for Indian martial arts as the origin of Asian (unarmed) martial arts, independent of any claims of transmission by Bodhidharma to Shaolin, there simply isn't any evidence. Indian martial arts aren't the oldest unarmed martial arts we have any evidence of - that honour goes to Mesopotamia, and this is most likely simply a case that the Mesopotamian evidence is better preserved (and there is older Egyptian evidence of unarmed martial arts). It is very likely that unarmed martial arts - notably, wrestling - existed long before any of the surviving evidence, and long, long before kalaripayattu, despite the claims some make of it being the mother of all martial arts.
There was, however, some transmission of military techniques and technology from the west to China. Notably, when the chariot was developed, probably in Central Asia, there was a relatively rapid spread of this technology to the east (China), the south (India) and the west (Europe), accompanied by speakers of Indo-European languages. Chariotry, including chariot archery, would have been introduced along with the chariot. Later, cavalry was probably introduced into China as well (rather than being independently developed there), and horsemanship and mounted archery would have been more introduced martial arts. However, these weren't of Indian origin.
On the early history of armed and unarmed martial arts: