Borel-Tukey morphisms between Ramsey homogeneity relations

We study Borel-Tukey morphisms between almost-homogeneity relations. For finite $r,s\ge 1$ and $m,n\ge 2$, we give the complete one-stage classification: ${\mathbf{H}\mathbf{o}\mathbf{m}}_m^r{\to }_{\mathrm{B}\mathrm{T}}{\mathbf{H}\mathbf{o}\mathbf{m}}_n^s$ holds exactly when $r>s$, or when $r=s$ and $m\ge n$. Our main result is that binary triple homogeneity requires exactly three sequential binary pair stages: ${\mathbf{H}\mathbf{o}\mathbf{m}}_2^2;{\mathbf{H}\mathbf{o}\mathbf{m}}_2^2;{\mathbf{H}\mathbf{o}\mathbf{m}}_2^2{⟶}_{\mathrm{B}\mathrm{T}}{\mathbf{H}\mathbf{o}\mathbf{m}}_2^3$ and ${\mathbf{H}\mathbf{o}\mathbf{m}}_2^2;{\mathbf{H}\mathbf{o}\mathbf{m}}_2^2\mathrm{̸}{⟶}_{\mathrm{B}\mathrm{T}}{\mathbf{H}\mathbf{o}\mathbf{m}}_2^3$. We also construct Borel morphisms ${\mathbf{H}\mathbf{o}\mathbf{m}}_2^2;{\mathbf{H}\mathbf{o}\mathbf{m}}_3^2{⟶}_{\mathrm{B}\mathrm{T}}{\mathbf{H}\mathbf{o}\mathbf{m}}_2^3$ and ${\mathbf{H}\mathbf{o}\mathbf{m}}_2^2;{\mathbf{H}\mathbf{o}\mathbf{m}}_2^2{⟶}_{\mathrm{B}\mathrm{T}}{\mathbf{H}\mathbf{o}\mathbf{m}}_k^2$ for every finite positive $k$.