  We show that the symmetric simple exclusion process (SSEP) on an 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 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. 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.
