  We prove some new statements on the distribution of $\mathcal{B}$-free
numbers in short intervals. In particular, we show an asymptotic result for the
variance of the number of $\mathcal{B}$-free integers in random short intervals
which are, in some sense, uniformly distributed.
  We establish a connection between our work and the paper by El Abdalaoui,
Lema\'nczyk & de la Rue on a flow associated to $\mathcal{B}$-free integers. In
addition, we study an analog of our variance for $k$-free integers in a number
field which provides new information for the corresponding dynamical system
constructed by Cellarosi & Vinogradov.
