There were in fact several encounters between Fransiscan friars and Buddhists. Whether it resulted in transfer of "mysticism" can be questioned, but as with all religions, mixing and absorption of ideas is quite likely.
Willem van Ruysbroeck, a Flemish Franciscan friar, spent six months in 1254 in Karakorum. In his "Itinirarium" he describes accurately Tibetan lamas. He also quotes the chant "Om mani padme hum" (Ou man haetavi or On man baccam, Sinica Franciscana).
Pope Nicholas IV sent Friar John of Monte Corvino (1247-1328) to China. He arrived in Khanbaliq (Peking) in 1294. He lived for several years in China. From here he sent two letters, the first dated 8 January 1305, the second 13 February 1306, in which he mentions Buddhists.
In 1328 the Franciscan friar Odoric de Pordenone arrived in Peking. In 1330 he returned to Padua where he dictated the story of his travels which resulted in the book "Relatio". The last papal envoy is John Marignolli who was sent to China in 1339 by Pope Benedict XII. He arrived in Khanbaliq in 1342 where he remained for three years. He returned in 1352 via Ceylon, another Buddhist country.
The travels of the friars aroused much interest in Europe. The most
popular work, which contains many legends apart from information
obtained from the writings of the friars, is John Mandeville's
"Voyager", written in 1365. There are about 300 manuscripts of this
work which was translated into most European languages and printed 22
times between ca. 1470 and the end of the eighteenth century.
Marco Polo's book has extensive mention and descriptions of Buddhists, which is in the same period, although no direct Fransiscan connection is documented in his work. But it seems clear that contact between Europe and the Orient was well established. Also, the Italian Franciscan friar John
of Pian di Carpino left Lyons in 1245 reaching Central Mongolia the following year. In 1247 he returned to France and wrote the "Tstoria Mongalorum". He mentions the religion of the Kitai. But Henri de Lubac asserts that these were Confucianists and not Buddhists that the friar met.
In later years Christianity (Missionaries) had innumerable well documented contacts with Buddhism, but these were (as far as i understand) not with Franciscan friars as such.