In the United States, one of the largest tensions between scientific research and religious beliefs has been around acceptance of evolution as an explanation for the diversity of modern life forms. Did this controversy exist in Hindu, Buddhist, or Shinto cultures, and if so, how did it (if ever) become accepted?
Darwinism and religion, even in the US, have not always been opposed. There have been periods when the two coexisted rather easily (e.g., the mid-19th c), and periods when religious interpretations of Darwinism have been more hostile (c. 1920s; 1980s, rise of the Christian right, I.D. and the "new" creationism). Historian Ron Numbers has written a lot about this during his career. See, for example, The Creationists (1992).
Now to your question...According to historian Osamu Sakura, there was little to no religious opposition to Darwinism in Meiji Japan (late-19th century). However, Sakura claims that most Japanese intellectuals interpreted Darwinism as having social consequences, and they were less concerned with the biological aspects. Hence, the character of evolutionary thought in Japan was more Spencerian than Darwinian.
Sakura, "Similarities and Varieties: A Brief Sketch on the Reception of Darwinism and Sociobiology in Japan," Biology and Philosophy 13 (1998): 341-357.
Many modern Hindu gurus (the type that take selfies on their motorbikes), argue that evolution is written into the Dasha Avatara or ten avatars of Vishnu. They kind of have a point, this is the avatars in chronological order:
Matsya (The Fish): The first class of Vertebrates, and an animal that evolved in the water.
Kurma (The Tortoise): An amphibious animal, lives in both land and water.
Varaha (The Boar): A wild beast, untamed, lives completely on the land.
Narasimha (The half-lion/half-man): Half man half beast, representing the first stage of man's evolution. He also has one of my favourite stories and kicks major ass (just saying).
Vamana (The Dwarf, or Incomplete Man): He is generally portrayed as an incomplete man, or a man in the process of growth by proponents of this view.
Parashurama (The Axe-Wielding Warrior): A base man, who used primitive weapons and existed only to fight and rule.
Rama (The King): Rama is a creator of a community, and is thus seen as an indication of humans creating communal societies.
Krishna (The Socially Complex Human): Krishna is by far the most "complete" and relatable human, with complex intentions and desires, although he still ultimately has a good heart.
Buddha (The Enlightened Human): He is seen as humanity gaining knowledge and understanding of themselves and the world around them.
Kalki (The Destructive Man): The one who is supposed to end human civilisation, Kalki is seen as a human with advanced weaponry capable of destroying the world.
These aren't actually huge leaps from the actual scripture...just a slightly more coloured reading of our scriptures.