Catholic missionaries to Tibet in the early part of the last century were struck by the outward resemblances to Catholic liturgy and discipline that were presented by Lamaismâ€”its infallible head, grades of clergy corresponding to bishop and priest, the cross, mitre, dalmatic, cope, censer, holy water, etc. At once voices were raised proclaiming the Lamaistic origin of Catholic rites and practices. Unfortunately for this shallow theory, the Catholic Church was shown to have possessed these features in common with the Christian Oriental churches long before Lamaism was in existence. The wide propagation of Nestorianism over Central and Eastern Asia as early as A.D. 635 offers a natural explanation for such resemblances as are accretions on Indian Buddhism.
Buddhism, Catholic Encyclopedia
via Daniel Mitsui
It seems to me there are pop-speculations aplenty about Buddhistic influences on Christianity, but little about influences in the reverse direction. I do not know whether present comparative religion studies would back up this article from the early twentieth century. However, Peter Leithart acting in a similar spirit once suggested that influences from liberal Protestantism helped sift Buddhism into a Sunday-school friendly form of religion.