We describe basics of a new approach to transverse momentum dependence | in hard exclusive processes. We develop it in application to the transition process gamma* gamma -> pi^0 at the handbag level. Our starting point is coordinate representation for matrix elements of operators describing a hadron with momentum p. Treated as functions of pz and z^2, they are parametrized through virtuality distribution amplitudes (VDA) \Phi (x, \sigma), with | x being Fourier-conjugate to (pz) and \sigma Laplace-conjugate to z^2. | For intervals with z^+=0, we introduce the transverse momentum distribution amplitude (TMDA) Psi (x, k_\perp), and write it in terms of VDA \Phi (x,\sigma). |The results of covariant calculations, written in terms of Phi (x, \sigma) are converted into expressions involving Psi (x, k_\perp). Starting with scalar toy models, we extend the analysis onto the case of spin-1/2 quarks and QCD. | We propose simple models for soft VDAs/TMDAs, and use them for comparison of handbag results with experimental (BaBar and BELLE) data on the pion transition form factor. We also discuss how one can generate high-k\perp tails from primordial soft distributions.