Barium zirconium sulfide (BaZrS3) is an earth-abundant and environmentally friendly chalcogenide perovskite with promising properties for various energy conversion applications. Recently, sulfurization of oxide precursors has been suggested as a viable solution for effective synthesis, especially from the perspective of circumventing the difficulty of handling alkali earth metals. In this work, we explore in detail the synthesis of BaZrS3 from Ba-Zr-O oxide precursor films sulfurized at temperatures ranging from 700 degrees C to 1000 degrees C. We propose a formation mechanism of BaZrS3 based on a two-step reaction involving an intermediate amorphization step of the BaZrO3 crystalline phase. We show how the diffusion of sulfur (S) species in the film is the rate-limiting step of this reaction. The processing temperature plays a key role in determining the total fraction of conversion from oxide to sulfide phase at a constant flow rate of the sulfur-containing H2S gas used as a reactant. Finally, we observe the formation of stoichiometric BaZrS3 (1:1:3), even under Zr-rich precursor conditions, with the formation of ZrO2 as a secondary phase. This marks BaZrS3 quite unique among the other types of chalcogenides, such as chalcopyrites and kesterites, which can instead accommodate quite a large range of non-stoichiometric compositions. This work opens up a pathway for further optimization of the BaZrS3 synthesis process, straightening the route towards future applications of this material.