Studies of model microswimmers have significantly contributed to the understanding of the principles of self-propulsion we have today. However, only a small number of microswimmer types have been amenable to analytic modeling, and further development of such approaches is necessary to identify the key features of these active systems. Here we present a general perturbative calculation scheme for swimmers composed of beads interacting by harmonic potentials and via hydrodynamics, driven by an arbitrary force protocol. The approach can be used with mobility matrices of arbitrary accuracy, and we illustrate it with the Oseen and Rotne-Prager approximations. We validate our approach by using 3 bead assemblies and comparing the results with the numerically obtained full-solutions of the governing equations of motion, as well as with existing analytic models for the linear and the triangular swimmer geometry. While recovering the relation between the force and swimming velocity, our detailed analysis and the controlled level of approximation allow us to find qualitative differences already in the far field flow of the devices. Consequently, we are able to identify a behavior of the swimmer that is richer than predicted in previous models. Given its generality, the framework can be applied to any swimmer geometry, driving protocol and bead interactions, as well as in problems involving many swimmers.Despite its immense usefulness, this model suffers from the constriction of all internal degrees of freedom by the swimming stroke, namely the internal dynamical behavior of the swimmer cannot react to its surrounding. This was overcome by replacing the arms with springs and prescribing the forces acting on the beads rather than the stroke itself [28,20]. Importantly, the responsive elastic degree of freedom, which is typically associated with the swimmer design, but which could also be in the fluid, is responsible for several interesting phenomena. For example, it is the source of the optimal driving frequency in the overdamped regime [29] and it promotes swimming based on the minimization of drag or enhancement of hydrodynamic interactions [20]. Furthermore, it is also responsible for the existence of a viscosity maximising the swimming velocity [30] and synchronization effects of the stroke [20]. Similar findings have been reported in investigations of bead-based swimmers in a visco-elastic fluid [31,32]. Recently, an altered version of the bead-spring model has been proposed, where the swimmer was driven by periodic changes in the equilibrium lengths of the springs [33,34].The boundedness of the linear swimmer to one dimension is broken in a triangular swimmer geometry, allowing for translational as well as rotational motion [35,14,16]. This geometry has also been used to model Chlamydomonas reinhardtii and investigate in particular the synchronization between the beating of its two flagella [18,36,19]. Experimentally, a triangular swimmer with intrinsic elasticity has been realized by placing ferromagnetic beads subject...