Emerging applications in the Internet of Things (IoT) and edge computing/learning have sparked massive renewed interest in developing distributed versions of existing (centralized) iterative algorithms often used for optimization or machine learning purposes. While existing work in the literature exhibits similarities, for the tasks of both algorithm design and theoretical analysis, there is still no unified method or framework for accomplishing these tasks. This article develops such a general framework for distributing the execution of (centralized) iterative algorithms over networks in which the required information or data is partitioned between the nodes in the network. This article furthermore shows that the distributed iterative algorithm, which results from the proposed framework, retains the convergence properties (rate) of the original (centralized) iterative algorithm. In addition, this article applies the proposed general framework to several interesting example applications, obtaining results comparable to the state of the art for each such example, while greatly simplifying and generalizing their convergence analysis. These example applications reveal new results for distributed proximal versions of gradient descent, the heavy ball method, and Newton's method. For example, these results show that the dependence on the condition number for the convergence rate of this distributed heavy ball method is at least as good as that of centralized gradient descent.