The throttle linkage to the transmission does more than cause a downshift. It also regulates hydraulic pressure to the clutches and bands. Downshifting and upshifting are as much functions of the governor...operated by the rotational speed of the output shaft...as the position of the throttle lever.
You have a Weber carb...an excellent unit...but never factory-installed on a 6-cylinder Eagle. An aftermarket carb...combined with the perception of going into neutral on a hill...leads me to suspect slipping. Eaglefreek is right: something may be amiss in the linkage that operates the throttle lever on the transmission.
If the linkage turns out to be the fault it must be repaired or properly adjusted immediately. Slipping clutches and bands will quickly self-destruct.
One other note: transmission fluid level...either over-or under-full...will adversely affect transmission operation.
I wish you good luck. Please keep us posted.