This topic continues to be of great interest. The care sheet from McNeela specifally for the Murray flute advises not to oil the Murray Blackwood flute. But McNeela have a Blackwood flute of their own almost identical and the care instructions in the web page say:
Caring for your McNeela African Blackwood flute
Dry your flute thoroughly after playing to prevent trapped moisture distorting and warping the wood. Always take it apart after playing and when storing, this takes pressure off the tenons. Oil the bore regularly to prevent from drying out but be careful not to over-oil it. Use cork grease on the cork lapping to allow for ease of disassembling. Never pull when disassembling, rather twist each part backwards and forwards and gently coax it off. Try to avoid very dry conditions and do not leave the flute in direct sunlight or near a source of heat. Always warm up before play.
I cannot see that a Murray flute would be any different in this regard. Seems somewhat of a contradiction.
Anyway, I reckon light oiling is beneficial.
Amusingly, it's 100% humidity here today in Melbourne, for our Melbourne Cup Horse Race, which is a grand public holiday.