Let F be a complex foliation by Riemann surfaces defined by a trivial (in the differentiable sense) fibration π:M⟶B but for which the complex structure on each fibre π−1(t) may depend on t. Let σ:B⟶M be a section of π contained in a F-relatively compact subset of M. We prove: for any F-relatively compact open set U containing Σ=σ(B) and any integer s≥0, there exists a function U⟶C of class Cs nonconstant on any leaf of (U,F), meromorphic along the leaves and whose set of poles is exactly Σ.