In recent years, there has been a growing body of research dedicated to countering the harmful consequences of outliers on range-based source localization. In contrast, this paper focuses on a lesser-explored but equally significant scenario where the source is in motion. The primary objective of this research is to simultaneously estimate both the position and velocity of a moving source. The current approaches, which primarily utilize least squares (LS) techniques, heavily rely on relaxation and linearization methods. Unfortunately, these techniques often yield unsatisfactory estimated results. Moreover, these approaches exhibit deficiencies in properly handling outlier range measurements and often fail to accurately estimate both the location and velocity of the source. This paper addresses the issue of outliers in the simultaneous estimation of position and velocity. It devises a novel approach that employs a neurodynamic method based on the augmented Lagrange programming neural network (ALPNN) framework. Unlike non-outlier LS methods with relaxation techniques, the proposed ALPNN method successfully implements a least absolute deviation estimator. Through simulations, it has been demonstrated that our ALPNN-enabled algorithm effectively achieves robust estimation of the position and velocity of a mobile source.