  We investigate systems of the form $\{A^tg:g\in\mathcal{G},t\in[0,L]\}$ where
$A \in B(\mathcal{H})$ is a normal operator in a separable Hilbert space
$\mathcal{H}$, $\mathcal{G}\subset \mathcal{H}$ is a countable set, and $L$ is
a positive real number. Although the main goal of this work is to study the
frame properties of $\{A^tg:g\in\mathcal{G},t\in[0,L]\}$, as intermediate
steps, we explore the completeness and Bessel properties of such systems from a
theoretical perspective, which are of interest by themselves. Beside the
theoretical appeal of investigating such systems, their connections to
dynamical and mobile sampling make them fundamental for understanding and
solving several major problems in engineering and science.
