Ok! I finally got the thing cleaned-up!
I tried WD-40 like you said, it just sort of half worked.
When I looked closer, I noticed that a lot of the white crud that was stuck-on had a slightly soapy consistency... weird.
I think it was the plastic that is starting to degrade.
So, anyways, I used some elbow grease, a toothbrush and some window washing liquid and managed to get most of it off. I can't reach the insides but I figure it won't hurt.
For this version, they used a kind of soft plastic which reminds me a lot of the plastic used in cheap toys. I don't think it ages very well (contrary to lovely Bakelite). Still feels soapy to the touch though, but at least it's back to its original gray color.
I think mine was made somewhere between 1935 and 1960 by looking at the logo on the box. No wonder why the plastic has seen better days
Thanks for the tips