The Cadmium Zinc Telluride Imager on AstroSat has proven to be a very effective all-sky monitor in the hard X-ray regime, detecting over three hundred GRBs and putting highly competitive upper limits on X-ray emissions from gravitational wave sources and fast radio bursts. We present the algorithms used for searching for such transient sources in CZTI data, and for calculating upper limits in case of non-detections. We introduce CIFT: the CZTI Interface for Fast Transients, a framework used to streamline these processes. We present details of 88 new GRBs detected by this framework that were previously not detected in CZTI.