Almonaci, 

I think you'll need the compute_projection_from_face_quadrature_points_matrix 
tfunction. 
Here's how you might use it:

std::vector<LA::MPI::Vector> distributed_field;
distributed_field.reinit(dof_handler.locally_owned_dofs(), 
mpi_communicator);

FullMatrix<double> qpoint_to_dof_matrix(fe.dofs_per_face,
face_quadrature.size());

std::vector<Vector<double>> local_values_at_qpoints, local_fe_values;
local_values_at_qpoints.reinit(face_quadrature.size());
local_fe_values.reinit(fe.dofs_per_cell);

for (const auto cell :dof_handler.active_cell_iterators())
for (unsigned int face = 0; face < GeometryInfo<dim>::faces_per_cell; ++
face)
if(face_at_boundary)
{
FETools::compute_projection_from_face_quadrature_points_matrix(fe,
face_quadrature,
face_quadrature,
cell,
face,
qpoint_to_dof_matrix);

for (unsigned int q = 0; q < face_quadrature->size(); ++q)
{
local_values_at_qpoints[q] = kirchof_stress_or_something;
}

qpoint_to_dof_matrix.vmult_add(local_fe_values,
local_values_at_qpoints);


cell->set_dof_values(local_fe_values,
distributed_field);

local_fe_values = 0;
}


The  distributed_field should have what you want in the end. Hope this 
helps. 

Abbas 

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