A desirable trait of an artificial agent acting in the visual world is to continually learn a sequence of language-informed tasks while striking a balance between sufficiently specializing in each task and building a generalized knowledge for transfer. Selective specialization, i.e., a careful selection of model components to specialize in each task, is a strategy to provide control over this trade-off. However, the design of selection strategies requires insights on the role of each model component in learning rather specialized or generalizable representations, which poses a gap in current research. Thus, our aim with this work is to provide an extensive analysis of selection strategies for visually grounded continual language learning. Due to the lack of suitable benchmarks for this purpose, we introduce two novel diagnostic datasets that provide enough control and flexibility for a thorough model analysis. We assess various heuristics for module specialization strategies as well as quantifiable measures for two different types of model architectures. Finally, we design conceptually simple approaches based on our analysis that outperform common continual learning baselines. Our results demonstrate the need for further efforts towards better aligning continual learning algorithms with the learning behaviors of individual model parts.