Infiltration of a porous object is a common and very useful technique for strengthening, waterproofing, or otherwise altering shapes fabricated initially by a process that inherently produces a porous green body or other shape.
One difficulty is that you can't infiltrate with solids; the infiltrant needs to be liquid. In some cases it would be possible to infiltrate with a colloid or suspension, but once the solvent is removed, the shape becomes porous again.
Various ways of causing solid grains of material to expand into a foam are known: shooting rice out of steam cannons, heating waterglass (or vermiculite or perlite) to drive out the water, baking muffins to react their baking powder, and so on. If such processes are applicable to the infiltrant grains and produce a closed-cell foam, they can seal up the porosity.
Construction spray foam illustrates another approach to foaming via heating: the polymerization reaction of the foam produces heat which produces the foaming gas.