A bounded Hahn field is a directed union of Hahn subfields of some field of generalised power series. Examples are the various fields of transseries: grid-based transseries, logarithmic-exponential series, exponential-logarithmic series, $\kappa$-bounded series, surreal numbers. We consider strongly linear maps defined between certain linear subspaces of bounded Hahn fields. We give sufficient conditions on the kernel for such maps to admit strongly linear sections. On the other hand, we derive surjectivity from asymptotic surjectivity, exploiting the structure of bounded fields as unions of spherically complete fields. Our conditions apply to automorphisms and derivations on bounded Hahn fields. In particular, we show how to construct logarithms from strongly linear derivations. As byproduct, we build bounded Hahn fields with compatible logarithms and derivations which violate the condition called T4 in the thesis of Schmeling. This is in clear contrast to the above cited examples of fields of transseries. This is joint work with A. Berarducci, S. Kuhlmann, M. Matusinski.