  In this paper we develop a framework for multivariate functional
approximation by a suitable Gaussian process via an exchangeable pairs coupling
that satisfies a suitable approximate linear regression property, thereby
building on work by Barbour (1990) and Kasprzak (2020). We demonstrate the
applicability of our results by applying it to joint subgraph counts in an
Erd\H{o}s-Renyi random graph model on the one hand and to vectors of weighted,
degenerate $U$-processes on the other hand. As a concrete instance of the
latter class of examples, we provide a bound for the functional approximation
of a vector of success runs of different lengths by a suitable Gaussian process
which, even in the situation of just a single run, would be outside the scope
of the existing theory.
