Identifying meaningful latent representations within diffusion models remains a challenging problem for causal approaches. We propose Causal Manifold Transport Diffusion Model (CMT-Diff), a framework that operationalizes causal actions as geometric transformations. By adopting the perspective of backtracking counterfactuals, we formulate the generative process as a composite diffeomorphism that couples the Probability Flow ODE with a Continuous Normalizing Flow. This mapping constructs an exogenous manifold where causal factors align with coordinate variations. Within this geometry, we derive Causal Manifold Transport (CMT) to realize interventions as linear vector translations along factor-aligned directions. We establish theoretical identifiability guarantees and demonstrate that our approach facilitates controllable generation by capturing the underlying causal manifold.