  We prove the existence of functions $f$ in the Bloch space of the unit ball
$\mathbb{B}_N$ of $\mathbb{C}^N$ with the property that, given any measurable
function $\varphi$ on the unit sphere $\mathbb{S}_N$, there exists a sequence
$(r_n)_n$, $r_n\in (0,1)$, converging to $1$, such that for every $w\in
\mathbb{B}_N$, $$f(r_n(\zeta -w)+w) \to \varphi(\zeta)\text{ as }n\to
\infty\text{, for almost every }\zeta \in \mathbb{S}_N.$$ The set of such
functions is residual in the little Bloch space. A similar result is obtained
for the Bloch space of the polydisc.
