A collision model (CM) is a framework to describe open quantum dynamics. In
its {\it memoryless} version, it models the reservoir R as
consisting of a large collection of elementary ancillas: the dynamics of the
open system S results from successive "collisions" of S
with the ancillas of R. Here, we present a general formulation of
memoryless {\it composite} CMs, where S is partitioned into the very
open system under study S coupled to one or more auxiliary systems {Si}.
Their composite dynamics occurs through internal S-{Si} collisions
interspersed with external ones involving {Si} and the reservoir R. We show that important known instances of quantum {\it non-Markovian}
dynamics of S -- such as the emission of an atom into a reservoir featuring a
Lorentzian, or multi-Lorentzian, spectral density or a qubit subject to random
telegraph noise -- can be mapped on to such {\it memoryless} composite CMs.Comment: 12 pages, 4 figure