We show that if Γ is a discrete subgroup of the group of the isometries of H k , and if ρ is a representation of Γ into the group of the isometries of H n , then any ρ-equivariant map F : H k → H n extends to the boundary in a weak sense in the setting of Borel measures. As a consequence of this fact, we obtain an extension of a result of Besson, Courtois and Gallot about the existence of volume non-increasing, equivariant maps. Then, we show that the weak extension we obtain is actually a measurable ρ-equivariant map in the classical sense. We use this fact to obtain measurable versions of Cannon-Thurston-type results for equivariant Peano curves. For example, we prove that if Γ is of divergence type and ρ is non-elementary, then there exists a measurable map D : ∂H k → ∂H n conjugating the actions of Γ and ρ(Γ). Related applications are discussed. Contents 1. Introduction 1 2. Definitions, notation and preliminary facts 5 3. Construction of B-C-G natural maps 10 4. Weak extension of equivariant maps: existence of developing measures 13 5. Sequence of ε-natural maps 19 6. Rigidity of representations 22 7. Measurable extension of natural maps 23 8. Measurable Cannon-Thurston maps 28 9. Convergence of Cannon-Thurston maps 31 References 32