We study convexity preserving properties for a class of time-dependent Hamilton-Jacobi equations in complete geodesic spaces. Convexity preserving properties for nonlinear evolution equations are well known in the Euclidean space. We extend the classical results for first order equations to the Busemann spaces such as a junction by using a recently developed theory of viscosity solutions on geodesic spaces. This talk is based on a joint work with Qing Liu (Fukuoka University).