We construct a projection-based cluster-additive transformation that block-diagonalizes wide classes of lattice Hamiltonians H = H 0 + V . Its cluster additivity is an essential ingredient to set up perturbative or non-perturbative linked-cluster expansions for degenerate excitation subspaces of H 0 . Our transformation generalizes the minimal transformation known amongst others under the names Takahashi's transformation, Schrieffer-Wolff transformation, des Cloiseaux effective Hamiltonian, canonical van Vleck effective Hamiltonian or two-block orthogonalization method. The effective cluster-additive Hamiltonian and the transformation for a given subspace of H, that is adiabatically connected to the eigenspace of H 0 with eigenvalue e n 0 , solely depends on the eigenspaces of H connected to e m 0 with e m 0 ≤ e n 0 . In contrast, other cluster-additive transformations like the multi-block orthognalization method or perturbative continuous unitary transformations need a larger basis. This can be exploited to implement the transformation efficiently both perturbatively and non-perturbatively. As a benchmark, we perform perturbative and non-perturbative linked-cluster expansions in the low-field ordered phase of the transverse-field Ising model on the square lattice for single spin-flips and two spin-flip bound-states.