  Entanglement engines are autonomous quantum thermal machines designed to
generate entanglement from the presence of a particle current flowing through
the device. In this work, we investigate the functioning of a two-qubit
entanglement engine beyond the steady-state regime. Within a master equation
approach, we derive the time-dependent state, the particle current, as well as
the associated current correlation functions. Our findings establish a direct
connection between coherence and internal current, elucidating the existence of
a critical current that serves as an indicator for entanglement in the steady
state. We then apply our results to investigate kinetic uncertainty relations
(KURs) at finite times. We demonstrate that there are more than one possible
definitions for KURs at finite times. While the two definitions agree in the
steady-state regime, they lead to different parameter's ranges for violating
KUR at finite times.
