In the present work we explore the interaction of a quasi-one-dimensional line kink of the sine-Gordon equation moving in two-dimensional spatial domains. We develop an effective equation describing the kink motion, characterizing its center position dynamics as a function of the transverse variable. The relevant description is valid both in the Hamiltonian realm and in the nonconservative one bearing gain and loss. We subsequently examine a variety of different scenarios, without and with a spatially dependent heterogeneity. The latter is considered both to be one dimensional (y independent) and genuinely two dimensional. The spectral features and the dynamical interaction of the kink with the heterogeneity are considered and comparison with the effective quasi-one-dimensional description (characterizing the kink center as a function of the transverse variable) is also provided. Generally, good agreement is found between the analytical predictions and the computational findings in the different cases considered.