The longstanding debate around the trade-offs of formalizing institutions of citizen participation (ICPs) within legal frameworks is currently facing a revival with the spread of citizens' assemblies. Among arguments in favour of "more" institutionalization, the expectation that it will protect ICPs from eventual political changes stands out. Among arguments against institutionalization, the fear of crystallizing certain "recipes" translates concerns that overly routinized institutions can be more easily manipulated. But what exactly does institutionalization entail? We offer three contributions to this conversation. First, we define institutionalization, identifying four constitutive dimensions: formalization, political embedding, professionalization, and social engagement. It is rare for any ICP to achieve high degrees of all four dimensions. Second, we propose four connected arguments: (a) the debate is not whether to institutionalize or not but about contextual configurations, (b) formalization alone does not guarantee the persistence or success of ICPs, (c) the extent and form of institutionalization will and should vary across ICPs and over time, and (d) if any single dimension of institutionalization stands out as crucial, it is political embedding. Third, we explore these ideas by analyzing three Latin American ICPs: popular initiatives (direct decision-making), participatory budgeting (deliberative decision-making), and sortition-based citizens' assemblies (consultative deliberation).