We show that the symmetric simple exclusion process (SSEP) on a countable set is well defined by the stirring graphical construction as soon as the dynamics of a single particle is. The resulting process is Feller, its Markov generator is derived on local functions, duality at the level of the empirical density field holds. We also provide a general criterion assuring that local functions form a core for the generator. We then move to the simple exclusion process (SEP) and show that the graphical construction leads to a well defined Feller process under a percolation-type assumption corresponding to subcriticality in a percolation with random inhomogeneous parameters. We derive its Markov generator on local functions which, under an additional general assumption, form a core for the generator. We discuss applications of the above results to SSEPs and SEPs in random environments, where the standard assumptions to construct the process and investigate its basic properties (by the analytic approach or by graphical constructions) are typically violated. As detailed in [14], our results for SSEP also allow to extend the quenched hydrodynamic limit in path space obtained in [11] by removing Assumption (SEP) used in there.