'foster a tradition' is correct and usable in written English.
You can use it when referring to creating, supporting, or promoting a particular practice. For example, "The school has fostered a tradition of excellence in the arts for over forty years."
Former United States House Speaker Sam Rayburn, a Democrat from North Texas who served in the House from 1913 until his death in 1961, fostered a tradition of home-state unity in his delegation.
These measures could also foster a cooperative tradition that can address rising concerns over nuclear trafficking and terrorism, which require regional coordination to combat.
The officers, Board of Directors, and staff of UJA-Federation of New York mourn the passing of Doris Slatkin, beloved sister of Jean Steinberg, whose family has fostered a long tradition of philanthropic generosity, commitment, and devotion that has enabled us to enrich the Jewish community worldwide.
Communities instead try to foster a spirit of bayanihan, a tradition of helping out your neighbour.
MySears and MyKmart were built to foster a community around the long tradition of shopping at those retailers.
The rebellious reaction is to foster a resistance identity defending local history, traditions, and authentic cultures.
We want to foster a culture of worship authentic to each religious tradition, and also a culture of forging personal bonds across religious traditions.
Meanwhile, other factors, including the fast-changing makeup of Britain's 60 million population, with more than 10-million people with origins far removed from Britain's ancient traditions, have combined to foster a debate about whether the country should shift to a more modern and democratic way of choosing its head of state.
The play's elegiac view of pastoral life and vanished agricultural traditions seems ideally tailored to foster a sense of history and cultural identity in young audiences.
