The publications of Gaia DR2 and DR3 have brought major improvements in stellar astrometry and photometry, particularly regarding the description of the white dwarf sequence. Notably, Gaia DR2 enabled the detection of variability in white dwarfs based solely on averaged astrometric and photometric quantities, i.e. the astrometric 5 parameters (positions, proper motion, and parallax) and general photometry properties in the G, BP and RP bands (mean, standard deviation and number of measurements). We identify and classify variable white dwarfs using Gaia DR3 data and Zwicky Transient Facility DR23 observations. The objective is to construct a catalogue of pulsating white dwarf candidates with robust selection criteria. We define a new sample of candidate variable white dwarfs using Gaia DR3 astrometric and photometric data. We cross-match this sample with the ZTF DR23 catalogue and apply a multiband Lomb-Scargle periodogram analysis to detect periodic variability. We then use the OPTICS unsupervised clustering algorithm to to group and classify the confirmed periodic stars. We identify 1423 variable white dwarfs candidates from Gaia DR3, with 864 having ZTF time series. 141 present significant periodicity. We classify these objects into known categories, including ZZ Ceti stars, GW Vir, V777 Her, and white dwarf-main sequence binaries. Our analysis yields several periodic stars, including three ZZ Ceti, 15 GW Vir, one V777 Her, and 24 WD-MS binaries. Furthermore, it reveals a significant population of potentialy variable stars, though without confirmed periodicity. Finally we publish our catalogue of candidate variable white dwarfs including variability status, periodicity, and classification information for the 864 sources with ZTF time series, 519 of them newly identified (including 83 new periodic stars).