It didn't feel awkward, probably because my approach was sufficiently coarse-grained: I didn't rely on any specific "lines" or "routines" to artificially generate closeness, but only on the general principle that the conversations should become progressively more personal and intimate.

I wish I could remind myself of the value of this principle more often, however. I often end up having superficial interactions with people I'd like to know better simply because that's the default way I relate to others. Recently, I started experimenting with Anki, using one side of the card to describe the relevant situation (e.g. "I find myself in the company of someone I'd like to get to know better"), and the other side to describe what I should do in that situation ("I ask questions that are progressively more personal and intimate").