In Kipling's time, a bride being transported in a procession alongside her groom would almost certainly have been on her way to the family house of her husband. This is not a specific Hindu tradition, but is a custom across many of the various religions practised in India.
Indian weddings are known the world over for their colour and exuberance. Filled with ritual and celebration, they can last several days. While the bride in Kim will most likely have had an arranged marriage, today an increasing number of couples are opting for love marriages.