Being really old my first knowledge of Big
Bands were those of the 40s of which the numbers were endless. These were "dance bands" as compared to "performance bands" So when you talk Big Bands they are totally different as are the drummers who played in them. In a dance band the kick is the main time mechanism that the band cues on and the dancers react too. 99% of the time the kick plays all four beats in 4/4 or 3/4 time.
And that holds true for Foxtrots, Waltzes and Polkas.
The Waltz can vary as to Strauss, Old Time and Modern as regards the kick but dance music is dance music. Whereas in the performance Big Bands the root is usually Jazz and how many out there can dance to Basie, Ellington, Herman etc.. well you get the picture. In the performance band the time keeper is most often Bass and a rhythm Guitar while the Drums and Keys do fills, accents etc.
As for those bands I'm a huge (late) Basie fan. In my opinion his 1968 album "Straight Ahead" is the
greatest ever by a Performance Big Band ever but I'm sure you all have your favourites.