The upcoming Legacy Survey of Space and Time (LSST) at the Vera C. Rubin Observatory is expected to detect a few million transients per night, which will generate a live alert stream during the entire ten years of the survey. This stream 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 the anticipated data, machine learning (ML) algorithms will be paramount for this task. We present the infrastructure tests and classification methods developed within the Fink broker in preparation for LSST. This work aims to provide detailed information regarding the underlying assumptions and methods behind each classifier and enable users to make informed follow-up decisions from 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 These include tree-based classifiers coupled with tailored feature extraction strategies as well as deep learning algorithms. Moreover, we introduce the CBPF (Centro Brasileiro de Pesquisas F\'isicas) Alert Transient Search (CATS), a deep learning architecture specifically designed for this task. Our results show that Fink classifiers are able to handle the extra complexity that is expected from LSST data. CATS achieved $ 93<!PCT!>$ precision for all classes except `long' (for which it achieved $ 83<!PCT!>$) while our best performing binary classifier achieves $ 98<!PCT!>$ precision and $ 99<!PCT!>$ completeness when classifying the periodic class. ELAsTiCC was an important milestone in preparing the Fink infrastructure to deal with LSST-like data. Our results demonstrate that Fink classifiers are well prepared for the arrival of the new stream, but this work also highlights that transitioning from the current infrastructures to Rubin will require significant adaptation of the currently available tools. This work was the first step in the right direction.