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In this paper we use generating function methods to obtain new asymptoticresults about spaces of $F$-stable maximal tori in $GL_n(\overline{F_q})$,$Sp_{2n}(\overline{F_q})$, and $SO_{2n+1}(\overline{F_q})$. We recoverstability results of Church--Ellenberg--Farb and Jim\'enez Rolland--Wilson for"polynomial" statistics on these spaces, and we compute explicit formulas fortheir stable values. We derive a double generating function for the charactersof the cohomology of flag varieties in type B/C, which we use to obtain analogsin type B/C of results of Chen: we recover "twisted homological stability" forthe spaces of maximal tori in $Sp_{2n}(\mathbb{C})$ and$SO_{2n+1}(\mathbb{C})$, and we compute a generating function for their "stabletwisted Betti numbers". We also give a new proof of a result of Lehrer usingsymmetric function theory.

Generating functions and statistics on spaces of maximal tori in classical Lie groups