Tumor migration within the microenvironment is a crucial aspect of cancer progression. Several external stimuli have been observed to influence and drive cell movement and, among them, hypoxia, i.e., the condition of inadequate oxygen supply, has a significant role, acting as an environmental stressor and inducing migratory bias and phenotypic changes. Here, we propose a mathematical framework for the description of tumor invasion and progression into the healthy tissue in response to different microenvironmental factors. Cells dynamics are modeled with a multilevel description that starts from the individual level, accounting for single-cell dynamics, and integrates them into a mesoscopic level, whose corresponding macroscopic limit leads to the evolutionary equation for the macroscopic cell density. The derived setting is numerically then tested in several scenarios with the main aim of showing tumor response to different microenvironmental changes.