  As a consequence of superradiant instability induced in Kerr black holes,
ultra-light boson clouds can be a source of persistent gravitational waves,
potentially detectable by current and future gravitational-wave detectors.
These signals have been predicted to be nearly monochromatic, with a small
steady frequency increase (spin-up), but given the several assumptions and
simplifications done at theoretical level, it is wise to consider, from the
data analysis point of view, a broader class of gravitational signals in which
the phase (or the frequency) slightly wander in time. Also other types of
sources, e.g. neutron stars in which a torque balance equilibrium exists
between matter accretion and emission of persistent gravitational waves, would
fit in this category. In this paper we present a robust and computationally
cheap analysis pipeline devoted to the search of such kind of signals. We
provide a full characterization of the method, through both a theoretical
sensitivity estimation and through the analysis of syntethic data in which
simulated signals have been injected. The search setup for both all-sky
searches and higher sensitivity directed searches is discussed.
