Conversational search is a relatively young area of research that aims at automating an information-seeking dialogue. In this paper we help to position it with respect to other research areas within conversational Artificial Intelligence (AI) by analysing the structural properties of an information-seeking dialogue. To this end, we perform a large-scale dialogue analysis of more than 150K transcripts from 16 publicly available dialogue datasets. These datasets were collected to inform different dialogue-based tasks including conversational search. We extract different patterns of mixed initiative from these dialogue transcripts and use them to compare dialogues of different types. Moreover, we contrast the patterns found in information-seeking dialogues that are being used for research purposes with the patterns found in virtual reference interviews that were conducted by professional librarians. The insights we provide (1) establish close relations between conversational search and other conversational AI tasks; and (2) uncover limitations of existing conversational datasets to inform future data collection tasks.