We extend the continuum theory of random intermittent search processes to the case of
N
independent searchers looking to deliver cargo to a single hidden target located somewhere on a semi-infinite track. Each searcher randomly switches between a stationary state and either a leftward or rightward constant velocity state. We assume that all of the particles start at one end of the track and realize sample trajectories independently generated from the same underlying stochastic process. The hidden target is treated as a partially absorbing trap in which a particle can only detect the target and deliver its cargo if it is stationary and within range of the target; the particle is removed from the system after delivering its cargo. As a further generalization of previous models, we assume that up to
n
successive particles can find the target and deliver its cargo. Assuming that the rate of target detection scales as
1
/
N
, we show that there exists a well-defined mean-field limit
N
→
∞
, in which the stochastic model reduces to a deterministic system of linear reaction-hyperbolic equations for the concentrations of particles in each of the internal states. These equations decouple from the stochastic process associated with filling the target with cargo. The latter can be modeled as a Poisson process in which the time-dependent rate of filling
λ
(
t
)
depends on the concentration of stationary particles within the target domain. Hence, we refer to the target as a Poisson trap. We analyze the efficiency of filling the Poisson trap with
n
particles in terms of the waiting time density
f
n
(
t
)
. The latter is determined by the integrated Poisson rate
μ
(
t
)
=
∫
t
0
λ
(
s
)
d
s
, which in turn depends on the solution to the reaction-hyperbolic equations. We obtain an approximate solution for the particle concentrations by reducing the system of reaction-hyperbolic equations to a scalar advection-diffusion equation using a quasisteady-state analysis. We compare our analytical results for the mean-field model with Monte Carlo simulations for finite
N
. We thus determine how the mean first passage time (MFPT) for filling the target depends on
N
and
n
.