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This work deals with the existence and uniqueness of μ -pseudo almost periodic solutions to some transport processes along the edges of a finite network with inhomogeneous conditions in the vertices. For that, the strategy consists of seeing these systems as a particular case of the semilinear boundary evolution equations (S H B E) du dt = A m u (t) + f (t , u (t)) , t ∈ R , L u (t) = g (t , u (t)) , t ∈ R , where A : = A m | k e r L generates a C 0 -semigroup admitting an exponential dichotomy on a Banach space. Assuming that the forcing terms taking values in a state space and in a boundary space respectively are only μ -pseudo almost periodic in the sense of Stepanov, we show that (SHBE) has a unique μ -pseudo almost periodic solution which satisfies a variation of constant formula. Then we apply the previous result to obtain the existence and uniqueness of μ -pseudo almost periodic solution to our model of network. [ABSTRACT FROM AUTHOR]