A great way to practice reading jazz big band figures is to write out pages and pages, and then read them at various tempos. The first thing I would work on, though, would be to make sure you can sing the figures like a horn player would, or in similar fashion to a jazz scat vocalist (Sarah Vaughn and Ella Fitzgerald come to mind). If you sing them like drum figures (short and staccato) they won’t swing!
Here are some examples. When I work on them, I sing the figures as swing rhythms (even though you might not know the specific pitches), and just fill as though in a big band. And listen to the big band greats such as Mel Lewis, Buddy Rich, Harold Jones, Jeff Hamilton, Peter Erskine, Ed Shaughnessy, John Riley, and Steve Fidyk for reference.
Danny Gottlieb has played with Pat Metheny, John McLaughlin, the NDR Big Band, and is currently a member of Gary Sinise’s Lt Dan Band (ltdanband.com). He is an associate professor of jazz studies at the University Of North Florida. dannygottlieb.org