Porous materials are highly relevant in engineering and medical applications due to their enhanced properties and lightweight nature. Current micromechanical models of porous materials can accurately predict the response under the assumptions of small deformations and drained conditions, typically driven by imposed deformations. However, the theoretical framework for the micromechanical modeling of porous material driven by pore pressure in the large-deformation range has been understudied. In this work, we develop a finite-deformation variational framework for pressure-driven foams, i.e., materials where the pore pressure in the cavities produces the deformation. We further consider different kinematical constraints in the formulation of boundary conditions: kinematic uniform displacements, periodic displacements and uniform traction. We apply the proposed model in the numerical simulation of lung porous tissue using a spherical alveolar geometry and an image-based geometry obtained from micro-computed-tomography images of rat lung. Our results show that the stress distributions in the spherical alveolar model are highly dependent on the kinematical constraints. In contrast, the stress distribution in the image-based alveolar model is not affected by the choice of boundary conditions. Further, when comparing the response of pressure-driven versus deformation-driven models, we conclude that hydrostatic stresses experience a marked shift in their distribution, whereas the deviatoric stresses remain unaffected. Our findings of how stresses are affected by the choice of boundary conditions and geometry take particular relevance in the simulation of the lungs, where the pressure-driven and deformation-driven cases are related to mechanical ventilation and spontaneous breathing.