Closed-loop decoder adaptation (CLDA) is an emerging paradigm for achieving rapid performance improvements in online brain-machine interface (BMI) operation. Designing an effective CLDA algorithm requires making multiple important decisions, including choosing the timescale of adaptation, selecting which decoder parameters to adapt, crafting the corresponding update rules, and designing CLDA parameters. These design choices, combined with the specific settings of CLDA parameters, will directly affect the algorithm's ability to make decoder parameters converge to values that optimize performance. In this article, we present a general framework for the design and analysis of CLDA algorithms and support our results with experimental data of two monkeys performing a BMI task. First, we analyze and compare existing CLDA algorithms to highlight the importance of four critical design elements: the adaptation timescale, selective parameter adaptation, smooth decoder updates, and intuitive CLDA parameters. Second, we introduce mathematical convergence analysis using measures such as mean-squared error and KL divergence as a useful paradigm for evaluating the convergence properties of a prototype CLDA algorithm before experimental testing. By applying these measures to an existing CLDA algorithm, we demonstrate that our convergence analysis is an effective analytical tool that can ultimately inform and improve the design of CLDA algorithms.