Abstract Detection of moving sources over a complicated background is important for several reasons. First is measuring the astrophysical motion of the source. Second is that such motion resulting from atmospheric scintillation, color refraction, or astrophysical reasons is a major source of false alarms for image-subtraction methods. We extend the Zackay, Ofek, and Gal-Yam image-subtraction formalism to deal with moving sources. The new method, named the translient (translational transient) detector, applies hypothesis testing between the hypothesis that the source is stationary and that the source is moving. It can be used to detect source motion or to distinguish between stellar variability and motion. For moving source detection, we show the superiority of translient over the proper image subtraction, using the improvement in the receiver-operating characteristic curve. We show that in the small translation limit, translient is an optimal detector of point-source motion in any direction. Furthermore, it is numerically stable, fast to calculate, and presented in a closed form. Efficient transient detection requires both the proper image-subtraction statistics and the translient statistics: When the translient statistic is higher, then the subtraction residual is likely due to motion. We test our algorithm both on simulated data and on real images obtained by the Large Array Survey Telescope. We demonstrate the ability of translient to distinguish between motion and variability, which has the potential to reduce the number of false alarms in transients detection. We provide the translient implementation in Python and MATLAB.