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Aims. We present a variability-, color-, and morphology-based classifier designed to identify multiple classes of transients and persistently variable and non-variable sources from the Zwicky Transient Facility (ZTF) Data Release 11 (DR11) light curves of extended and point sources. The main motivation to develop this model was to identify active galactic nuclei (AGN) at different redshift ranges to be observed by the 4MOST Chilean AGN/Galaxy Evolution Survey (ChANGES). That being said, it also serves as a more general time-domain astronomy study. Methods. The model uses nine colors computed from CatWISE and Pan-STARRS1 (PS1), a morphology score from PS1, and 61 single-band variability features computed from the ZTF DR11 g and r light curves. We trained two versions of the model, one for each ZTF band, since ZTF DR11 treats the light curves observed in a particular combination of field, filter, and charge-coupled device (CCD) quadrant independently. We used a hierarchical local classifier per parent node approach-where each node is composed of a balanced random forest model. We adopted a taxonomy with 17 classes: non-variable stars, non-variable galaxies, three transients (SNIa, SN-other, and CV/Nova), five classes of stochastic variables (lowz-AGN, midz-AGN, highz-AGN, Blazar, and YSO), and seven classes of periodic variables (LPV, EA, EB/EW, DSCT, RRL, CEP, and Periodic-other). Results. The macro-averaged precision, recall, and F1-score are 0.61, 0.75, and 0.62 for the g -band model, and 0.60, 0.74, and 0.61, for the r -band model. When grouping the four AGN classes (lowz-AGN, midz-AGN, highz-AGN, and Blazar) into one single class, its precision-recall, and F1-score are 1.00, 0.95, and 0.97, respectively, for both the g and r bands. This demonstrates the good performance of the model in classifying AGN candidates. We applied the model to all the sources in the ZTF/4MOST overlapping sky (−28 ≤ Dec ≤ 8.5), avoiding ZTF fields that cover the Galactic bulge (| gal_b | ≤ 9 and gal_l ≤ 50). This area includes 86 576 577 light curves in the g band and 140 409 824 in the r band with 20 or more observations and with an average magnitude in the corresponding band lower than 20.5. Only 0.73% of the g -band light curves and 2.62% of the r -band light curves were classified as stochastic, periodic, or transient with high probability ( P init ≥ 0.9). Even though the metrics obtained for the two models are similar, we find that, in general, more reliable results are obtained when using the g -band model. With it, we identified 384 242 AGN candidates (including low-, mid-, and high-redshift AGN and Blazars), 287 156 of which have P init ≥ 0.9.