The present study investigates the transient processes controlling ignition by a hot jet issued from a pre-chamber. Direct numerical simulations (DNS) have been performed to study the characteristics of the turbulent jet flow and of the associated flame during the whole ignition process, quantifying the relevant physicochemical interactions between pre-chamber and main chamber. Thanks to a detailed analysis of the DNS results, the transient ignition is found to consist of three main sequential processes: (1) near-orifice local ignition in the main chamber; (2) further flame development supported by the jet flow; and (3) global ignition and propagation of a self-sustained flame in the main chamber, independently from the hot jet. The characteristic time-scale of the hot jet as well as jet-induced effects (local enrichment, supply of radicals and heat) are found to be essential for successful ignition in the main chamber. A more intense turbulence in the main chamber appears to support local ignition. However, it also induces local quenching, thus delaying global ignition. An ignition threshold based on a critical Damköhler number is a promising concept, but is not sufficient to describe the process in all its complexity.