I don’t think there’s a ‘supposed to’ when it comes to structuring our work process as long as the job is done and it fulfils the purpose. People get things done in a variety of ways according to how it suits them best considering the circumstances.
I, for one, answer all the important and urgent emails directly. Yet, the ones which I should answer but don’t feel like answering at a given moment usually wait until the end of the night to receive one — although I would’ve read them partially or in their entirety at the given moment. These ones might even be important but not displaying a sense of urgency.
Other people could be more systematic in refusing to even look at their emails during the day because it might distract them from the work they have at hand. While others feel the need to answer their emails immediately because they prefer an immediately clean, well-organised inbox.
Bottom line, there’s no ‘supposed to.’