  Let $f$ be a positive multiplicative function and let $k\geq 2$ be an
integer. We prove that if the prime values $f(p)$ converge to $1$ sufficiently
slowly as $p\rightarrow +\infty$, in the sense that $\sum_{p}|f(p)-1|=\infty$,
there exists a real number $c>0$ such that the $k$-tuples
$(f(p+1),\ldots,f(p+k))$ are dense in the hypercube $[0,c]^k$ or in
$[c,+\infty)^k$. In particular, the values $f(p+1),\ldots,f(p+k)$ can be put in
any increasing order infinitely often. Our work generalises previous results of
De Koninck and Luca.
