What are examples of Indian cuisine influenced by the food culture of a new city or a new area?

Nearly all ethnic cuisines are affected when they move into a new region with its own food culture. Original ingredients may not be available and substitutes must be found. Local tastes may differ considerably and so lead to changes in the originals. Take the example of 'curry', itself a pretty wide category. One can find 'curry' in nearly every country in the world, but both the ingredient lists and the final product will vary significantly. A German Currywurst will taste nothing like a curry from Punjab which will taste nothing like one from Chennai. A British or American curried chicken will not be similar to the same product in Jamaica or Capetown. Foods that make heavy use of chili peppers, for instance, may be 'watered down' when they hit cultures that do not appreciate spicy hot foods.

The question sounds like it is about how was Indian cuisine adapted to a new city or region, in terms of ingredients, moderation of spices etc. If the question is taken the other way around - most Indian cuisine known in other parts of the world is not actually Indian. As and others have explained, cuisine of other regions have had a deep impact - to the point of assimilation - on Indian cuisine. Central Asian / Middle Eastern / Persian (most of the 'Tandoori' stuff, Pulao, Biryani), Tibetan, Chinese, Burmese, Portugese, South American (where do you think chillies, tomatoes and potatoes came to India from?), etc. cuisines are so deeply assimilated they are Indian. True Indian cuisine - plain, simple vegetarian and a few meat and fish dishes - is surprisingly available in many parts of the world, in a form very identifiable with India. For example, * Kare Raisu in Japan ('Curry Rice'; from South Indian gravy based dishes cooked by Indian sailors who visited Japan) * Roti Canai (Canai = Chana, or chickpeas / garbanzo) in Malasia * Curry Goat in the Caribbean * Sada roti in Trinidad & Tobago - which is Indian roti served for breakfast with a variety of vegetable dishes * 'Double' - or 2 poories with a chana filling - in the Caribbean * Phulouries & kachories in the Caribbean * Mulligatawny Soup (from the humble Indian rasam / mologatanni 'pepper water') imported from India to UK * Kedgeree (from the humble Indian Khichdi) adapted in UK * Vindaloo (usually pork; this itself is an adaptation of the Portugese Vin d'alho - wine / vinegar and garlic) adapted in a milder from in UK Then there is the distinct influence of Indian cuisine on Singaporean cuisine - Fish Head Curry, for example.