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We present a general framework to study the distribution of the flux through the origin up to time $t$, in a non-interacting one-dimensional system of particles with a step initial condition with a fixed density $ρ$ of particles to the left of the origin. We focus principally on two cases: (i) when the particles undergo diffusive dynamics (passive case) and (ii) run-and-tumble dynamics for each particle (active case). In analogy with disordered systems, we consider the flux distribution both for the annealed and the quenched initial conditions, for the passive and active particles. In the annealed case, we show that, for arbitrary particle dynamics, the flux distribution is a Poissonian with a mean $μ(t)$ that we compute exactly in terms of the Green's function of the single particle dynamics. For the quenched case, we show that, for the run-and-tumble dynamics, the quenched flux distribution takes an anomalous large deviation form at large times $P_{\rm qu}(Q,t) \sim \exp\left[-ρ\, v_0\, γ\, t^2 ψ_{\rm RTP}\left(\frac{Q}{ρv_0\,t} \right) \right]$, where $γ$ is the rate of tumbling and $v_0$ is the ballistic speed between two successive tumblings. In this paper, we compute the rate function $ψ_{\rm RTP}(q)$ and show that it is nontrivial. Our method also gives access to the probability of the rare event that, at time $t$, there is no particle to the right of the origin. For diffusive and run-and-tumble dynamics, we find that this probability decays with time as a stretched exponential, $\sim \exp(-c\, \sqrt{t})$ where the constant $c$ can be computed exactly. We verify our results for these large deviations by using an importance sampling Monte-Carlo method.