The upcoming Legacy Survey of Space and Time (LSST) at the Vera Rubin Observatory is expected to detect a few million transients per night, which will generate a live alert stream during the entire 10 years of the survey. This will be distributed via community brokers whose task is to select subsets of the stream and direct them to scientific communities. Given the volume and complexity of data, machine learning (ML) algorithms will be paramount for this task. We present the infrastructure tests and classification methods developed within the {\sc Fink} broker in preparation for LSST. This work aims to provide detailed information regarding the underlying assumptions, and methods, behind each classifier, enabling users to make informed follow-up decisions from {\sc Fink} photometric classifications. Using simulated data from the Extended LSST Astronomical Time-series Classification Challenge (ELAsTiCC), we showcase the performance of binary and multi-class ML classifiers available in {\sc Fink}. These include tree-based classifiers coupled with tailored feature extraction strategies, as well as deep learning algorithms. We introduce the CBPF Alert Transient Search (CATS), a deep learning architecture specifically designed for this task. Results show that {\sc Fink} classifiers are able to handle the extra complexity which is expected from LSST data. CATS achieved $97\%$ accuracy on a multi-class classification while our best performing binary classifier achieve $99\%$ when classifying the Periodic class. ELAsTiCC was an important milestone in preparing {\sc Fink} infrastructure to deal with LSST-like data. Our results demonstrate that {\sc Fink} classifiers are well prepared for the arrival of the new stream; this experience also highlights that transitioning from current infrastructures to Rubin will require significant adaptation of currently available tools.