  We present our follow-up observations with GRANDMA of transient sources
revealed by the Zwicky Transient Facility (ZTF). Over a period of six months,
all ZTF triggers were examined in real time by a dedicated science module
implemented in the Fink broker, which will be used for the data processing of
the Vera C. Rubin Observatory. In this article, we present three selection
methods to identify kilonova candidates. Out of more than 35 million
candidates, a hundred sources have passed our selection criteria. Six were then
followed-up by GRANDMA (by both professional and amateur astronomers). The
majority were finally classified either as asteroids or as supernovae events.
We mobilized 37 telescopes, bringing together a large sample of images, taken
under various conditions and quality. To complement the orphan kilonova
candidates (those without associated gamma-ray bursts, which were all), we
included three additional supernovae alerts to conduct further observations of
during summer 2021. We demonstrate the importance of the amateur astronomer
community that contributed images for scientific analyzes of new sources
discovered in a magnitude range r'=17-19 mag. We based our rapid kilonova
classification on the decay rate of the optical source that should exceed 0.3
mag/day. GRANDMA's follow-up determined the fading rate within 1.5+/-1.2 days
post-discovery, without waiting for further observations from ZTF. No confirmed
kilonovae were discovered during our observing campaign. This work will be
continued in the coming months in the view of preparing for kilonova searches
in the next gravitational-wave observing run O4.
