Consider a sequence of pointwise Lipchitzian operators \(\{H_k\}\) , and define a generalised iteration process as follows: starting from an arbitrary point \(u_1 \in C\) , generate a sequence \(\{u_k\}\) by iteratively computing \(u_{k+1} = H_k(u_k)\) for \(k \ge 1\) . If this process converges weakly to a common fixed point of a semigroup \(\mathcal {T}\) , regardless of the starting point, it serves as a method for approximating such common fixed points. In practical implementations, each iteration may introduce computational errors. Therefore, it is crucial to assess the stability of this process—specifically, whether it continues to converge weakly to a common fixed point when each iteration k produces a point \(x_{k+1}\) that is close to, but not exactly equal to \(H_k(x_k)\) . This chapter focuses on stability under summable errors, meaning that for any sequence \(\{x_k\}\) within C where \(\sum ^{\infty }_{k=1} \Vert x_{k+1} - H_k(x_k) \Vert < \infty ,\) the sequence converges in the weak topology to a common fixed point of \(\mathcal {T}\) , assuming that the sequence \(u_{k+1} = H_k(u_k)\) is weak-convergent to a (possibly different) common fixed point of \(\mathcal {T}\) for any initial element \(u_1 \in C\) . This chapter aims to present a general method for establishing a specific type of stability in iteration processes that converge to common fixed points of pointwise Lipschitzian semigroups. We illustrate this method using examples from the weak convergence results of generalised Krasnosel’skii-Mann, Ishikawa, and implicit iteration processes discussed in the previous chapters. These examples serve as patterns that can be adapted for various other contexts.