Right on, Leonard re/British Mullis v. Swiss Mullis. Alf Mullis of Austrailia compared modern Mullis families' residences (1500 a.d. to present) with Roger de Molis' Probate of a.d. 1295, and found modern Mullis families reside in localities, shires, etc. as identifiable on Roger's Probate.From 1066 Mullis has ranged from Meulles, Molis, Mulys, Moelis, Muelis, Moles and even Mules, until 16th century when it became Mullys, Mulles, Mullyes, and Mullis.
An International Registry of Mullis families compiled in 1995-96 revealed that of an estimated 5,085 households, fully 4,901 were in English-speaking nations, compared with 160 Swiss Mullis.Obviously Swiss Mullis folks traveled or migrated, but British Mullis origins predate Swiss Mullis by some three hundred years, so they did not originate in Switzerland.
An eye-opener on this subject is to check LDS's International Gen. Index for Mullis in UK, also Mulliss.