Given an $n$-dimensional random vector $X^{(n)}$ , for $k < n$, consider its $k$-dimensional projection $\mathbf{a}_{n,k}X^{(n)}$, where $\mathbf{a}_{n,k}$ is an $n \times k$-dimensional matrix belonging to the Stiefel manifold $\mathbb{V}_{n,k}$ of orthonormal $k$-frames in $\mathbb{R}^n$. For a class of sequences $\{X^{(n)}\}$ that includes the uniform distributions on scaled $\ell_p^n$ balls, $p \in (1,\infty]$, and product measures with sufficiently light tails, it is shown that the sequence of projected vectors $\{\mathbf{a}_{n,k}^\intercal X^{(n)}\}$ satisfies a large deviation principle whenever the empirical measures of the rows of $\sqrt{n} \mathbf{a}_{n,k}$ converge, as $n \rightarrow \infty$, to a probability measure on $\mathbb{R}^k$. In particular, when $\mathbf{A}_{n,k}$ is a random matrix drawn from the Haar measure on $\mathbb{V}_{n,k}$, this is shown to imply a large deviation principle for the sequence of random projections $\{\mathbf{A}_{n,k}^\intercal X^{(n)}\}$ in the quenched sense (that is, conditioned on almost sure realizations of $\{\mathbf{A}_{n,k}\}$). Moreover, a variational formula is obtained for the rate function of the large deviation principle for the annealed projections $\{\mathbf{A}_{n,k}^\intercal X^{(n)}\}$, which is expressed in terms of a family of quenched rate functions and a modified entropy term. A key step in this analysis is a large deviation principle for the sequence of empirical measures of rows of $\sqrt{n} \mathbf{A}_{n,k}$, which may be of independent interest. The study of multi-dimensional random projections of high-dimensional measures is of interest in asymptotic functional analysis, convex geometry and statistics. Prior results on quenched large deviations for random projections of $\ell_p^n$ balls have been essentially restricted to the one-dimensional setting.