  In this paper we show a functional central limit theorem for the sum of the
first $\lfloor t n \rfloor$ diagonal elements of $f(Z)$ as a function in $t$,
for $Z$ a random real symmetric or complex Hermitian $n\times n$ matrix. The
result holds for orthogonal or unitarily invariant distributions of $Z$, in the
cases when the linear eigenvalue statistic $\operatorname{tr} f(Z)$ satisfies a
CLT. The limit process interpolates between the fluctuations of individual
matrix elements as $f(Z)_{1,1}$ and of the linear eigenvalue statistic. It can
also be seen as a functional CLT for processes of randomly weighted measures.
