This question is rather broad and should be narrowed down to particular regions or countries, as Asia includes very heterogeneous countries. In the broadest terms, there is a strong increasing trend in foreign direct investment inflows in Asian countries, particularly after the 1990s. It has been strongly affected by the Global financial crisis, although there was a full recovery in the case of East-Asian economies and a much more unstable recovery in the case of South-Asian economies. It could be argued that in many of these countries, the investment was export-oriented and vertical, fueling the exports and economic growth of the host countries, and being a significant contributing factors their overall economic success (particularly in the case of Asian tigers). I would like to emphasize that this is a very broad generalization and that a country-level analysis would be much more meaningful.
I provide some data on general foreign direct investment trends for broad Asian regions: