Summary: Uncertainty principle significantly provides a bound to predict precision of measurement with regard to any two incompatible observables, and thereby plays a nontrivial role in quantum precision measurement. In this work, we observe the dynamical features of the quantum-memory-assisted entropic uncertainty relations (EUR) for a pair of incompatible measurements in an open system characterized by local generalized amplitude damping (GAD) noises. Herein, we derive the dynamical evolution of the entropic uncertainty with respect to the measurement affecting by the canonical GAD noises when particle \(A\) is initially entangled with quantum memory \(B\). Specifically, we examine the dynamics of EUR in the frame of three realistic scenarios: one case is that particle \(A\) is affected by environmental noise (GAD) while particle \(B\) as quantum memory is free from any noises, another case is that particle \(B\) is affected by the external noise while particle \(A\) is not, and the last case is that both of the particles suffer from the noises. By analytical methods, it turns out that the uncertainty is not full dependent of quantum correlation evolution of the composite system consisting of \(A\) and \(B\), but the minimal conditional entropy of the measured subsystem. Furthermore, we present a possible physical interpretation for the behavior of the uncertainty evolution by means of the mixedness of the observed system; we argue that the uncertainty might be dramatically correlated with the systematic mixedness. Furthermore, we put forward a simple and effective strategy to reduce the measuring uncertainty of interest upon quantum partially collapsed measurement. Therefore, our explorations might offer an insight into the dynamics of the entropic uncertainty relation in a realistic system, and be of importance to quantum precision measurement during quantum information processing.