  We show that there are an irrational rotation $Tx=x+\alpha$ on the circle
$\mathbb{T}$ and a continuous $\varphi\colon\mathbb{T}\to\mathbb{R}$ such that
for each (continuous) uniquely ergodic flow
$\mathcal{S}=(S_t)_{t\in\mathbb{R}}$ acting on a compact metric space $Y$, the
automorphism $T_{\varphi,\mathcal{S}}$ acting on $(X\times Y,\mu\otimes\nu)$ by
the formula $T_{\varphi,\mathcal{S}}(x,y)=(Tx,S_{\varphi(x)}(y))$, where $\mu$
stands for Lebesgue measure on $\mathbb{T}$ and $\nu$ denotes the unique
$\mathcal{S}$-invariant measure, has the property of asymptotically orthogonal
powers. This gives a class of relatively weakly mixing extensions of irrational
rotations for which Sarnak's conjecture on M\"obius disjointness holds for all
uniquely ergodic models of $T_{\varphi,\mathcal{S}}$. Moreover, we obtain a
class of "random" ergodic sequences $(c_n)\subset\mathbb{Z}$ such that if
$\boldsymbol{\mu}$ denotes the M\"obius function, then $$
\lim_{N\to\infty}\frac1N\sum_{n\leq N}g(S_{c_n}y)\boldsymbol{\mu}(n)=0 $$ for
all (continuous) uniquely ergodic flows $\mathcal{S}$, all $g\in C(Y)$ and
$y\in Y$.
