Singular cohomology of a limit of topological spaces

Question

Let $(X_\lambda)$ be a filtered projective system of topological spaces, $X = \varprojlim X_\lambda$ and let $R$ be a finite ring, for example $R = \mathbb{Z}/l$. Assuming that $(X_\lambda)$ is sufficiently nice (see below), is the a canoninal isomorphism of singular cohomology groups $$ H^n(X,R) \cong \varinjlim H^n(X_\lambda,R) \ ? $$ In my case, "sufficiently nice" has the following meaning. I consider a projective system of (possibly singular) complex varieties $(Y_\lambda)$ with finite transition morphisms and let $(X_\lambda) := (Y_\lambda(\mathbb{C}))$ be the corresponding system of complex spaces.