We study the interplay between two nontrivial boundary effects: (1) the two-dimensional (2d) edge states of three-dimensional (3d) strongly interacting bosonic symmetry-protected topological states, and (2) the boundary fluctuations of 3d bulk disorder-to-order phase transitions. We then generalize our study to 2d gapless states localized at an interface embedded in a 3d bulk, when the bulk undergoes a quantum phase transition. Our study is based on generic long-wavelength descriptions of these systems and controlled analytic calculations. Our results are summarized as follows: (i) The edge state of a prototype bosonic symmetry-protected state can be driven to a new fixed point by coupling to the boundary fluctuations of a bulk quantum phase transition; (ii) the states localized at a 2d interface of a 3dSU(N) quantum antiferromagnet may be driven to a new fixed point by coupling to the bulk quantum critical modes. Properties of the new fixed points identified are also studied. © 2020 American Physical Society.