Summary: We propose new concepts, a dynamically transversely trapping surface (DTTS) and a marginally DTTS, as indicators for a strong gravity region. A DTTS is defined as a two-dimensional closed surface on a spacelike hypersurface such that photons emitted from arbitrary points on it in transverse directions are acceleratedly contracted in time, and a marginally DTTS is reduced to the photon sphere in spherically symmetric cases. (Marginally) DTTSs have a close analogy with (marginally) trapped surfaces in many aspects. After preparing the method of solving for a marginally DTTS in the time-symmetric initial data and the momentarily stationary axisymmetric initial data, some examples of marginally DTTSs are numerically constructed for systems of two black holes in the Brill-Lindquist initial data and in the Majumdar-Papapetrou spacetimes. Furthermore, the area of a DTTS is proved to satisfy the Penrose-like inequality \(A_0\le 4\pi (3GM)^2\), under some assumptions. Differences and connections between a DTTS and the other two concepts proposed by us previously, a loosely trapped surface [T. Shiromizu et al., PTEP, Prog. Theor. Exper. Phys. 2017, No. 3, Article ID 033E01, 6 p. (2017; Zbl 1477.83009)] and a static/stationary transversely trapping surface [H. Yoshino et al., PTEP, Prog. Theor. Exper. Phys. 2017, No. 6, Article ID 063E01, 23 p. (2017; Zbl 1477.83070)], are also discussed. A (marginally) DTTS provides us with a theoretical tool to significantly advance our understanding of strong gravity fields. Also, since DTTSs are located outside the event horizon, they could possibly be related with future observations of strong gravity regions in dynamical evolutions.