Here are some possibilities:
Foreign universities may be interested in educating Indian citizens, but they'll come here primarily because they see a market here. [Here's an Indian analogy: If IIMs want to set up shop in the Middle East, is it because they feel that it's their mission to educate the citizens of those countries?]
They can use their Indian campus for the 'study abroad' programs for their own UG students. [In the case of IIMs, they can use their foreign campuses as an 'incentive' for their faculty -- by allowing them to spend some time in Dubai and Singapore.]
They may want to be known for their 'Global Campus,' however meaningless that label may be.
Instead of short term 'study abroad' program, the Indian campus may well turn into a destination for 'education tourism.'
If #1 -- the Indian market -- is the reason, I expect the for-profit universities (the University of Phoenix) to be interested, rather than the Harvards and the Stanfords. Even if the latter do come to India, it'll be to offer easy and cheap programs (that nevertheless are in high demand): business, economics, accounting and the like. Humanities are also easy and cheap to set up and run, and they help make the programs appear well-rounded. But, don't expect a program in the sciences or engineering -- except perhaps mathematics and computer science.
Can #2 -- Study Abroad progams -- be a good reason for an Indian campus? Frankly, I don't know. My guess is that American UG students are far more interested in Europe, Japan, Australia. India may be attractive to a few because it's exotic or 'emerging', but let's be realistic. Also, universities don't need their own campus for study abroad programs -- they can do it with a local partner.
#3 -- Global Campus -- should be recognized for what it really is: an empty slogan with perhaps some marketing value. It could, for example, help in getting a better place in THE-QS rank list. But here is the thing: Harvard and Stanford don't need this label; the University of Southeastern Idaho might.
#4 -- education tourism -- could be attractive for cost-conscious students from the US as well as elsewhere, if they can get an MIT degree at a lower cost made possible by the Indian operations. But would MIT want to be in this market?