  The problem of the detection statistics of a quantum walker has received
increasing interest, connected as it is to the problem of quantum search. We
investigate the effect of employing a moving detector, using a projective
measurement approach with fixed sampling time $\tau$, with the detector moving
right before every detection attempt. For a tight-binding quantum walk on the
line, the moving detector allows one to target a specific range of group
velocities of the walker, qualitatively modifying the behavior of the quantum
first-detection probabilities. We map the problem to that of a stationary
detector with a modified unitary evolution operator and use established methods
for the solution of that problem to study the first-detection statistics for a
moving detector on a finite ring and on an infinite 1D lattice. On the line,
the system exhibits a dynamical phase transition at a critical value of $\tau$,
from a state where detection decreases exponentially in time and the total
detection is very small, to a state with power-law decay and a significantly
higher probability to detect the particle. The exponent describing the
power-law decay of the detection probability at this critical $\tau$ is 10/3,
as opposed to 3 for every larger $\tau$. In addition, the moving detector
strongly modifies the Zeno effect.
