A 32 special was a standard cartridge like the .30-30.
You should find it in the same guns.
Winchester 94
Marlin 336 (36)
The .375 was in a "Big Bore" winchester 94 and the 375 Marlin (which is still a 336 based frame)
But no, I wouldn't call ether a big bore.
Not stirring it, nor am I calling it one, but Winchester sure called the .375 Win a Big Bore in there marketing scheme back a few years....
Marlin called it simply the Marlin 375.
I have the Marlin and the chunk 'o lead looks pretty damn big as does the bore...but I cannot and will not comment on what qualifies as a true Big Bore.