We prove tunneling estimates for two-dimensional Dirac systems which are localized in space due to the presence of a magnetic field. The Hamiltonian driving the motion admits the decomposition H=H0+W\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ H = H_0 + W$$\end{document}, where H0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_0 $$\end{document} is a rotationally symmetric magnetic Dirac operator and W is a position-dependent matrix-valued potential satisfying certain smoothness condition in the angular variable. A consequence of our results are upper bounds for the growth in time of the expected size of the system and its total angular momentum.