<p>In this work, we study the dynamics of a special type of transcendental meromorphic functions called shift-pseudoperiodic functions. It is proven that the Fatou sets of such functions are invariant under certain translations. We show that these functions can never have Baker wandering domains. Some non-existence criteria of Herman rings for such functions are given. A detailed discussion on completely invariant Fatou components is also provided. We further describe a class of shift-pseudoperiodic functions having wandering domains. Finally, a brief account of the dynamics of semigroups generated by shift-pseudoperiodic entire functions is presented.</p>