Beginning a conversation "from afar"

Alice wants to discuss with Bob something he likely does not want to discuss. So she first starts talking about something indirectly related to the topic, gradually approaching the topic, so that eventually they end up discussing it.

In Russian, one would say that she "began the conversation from afar" (начать беседу издалека), but this does not seem to be common usage in English.

Is there an English word or phrase describing such a way to open a conversation (by talking about something indirectly related or seemingly unrelated)?

EDIT: Lots of great answers, thanks!! Two clarifications:

1. The original Russian phrase does not have negative connotations (unlike, say, the English "Beating around the bush")
2. It can also apply to a monologue. E.g., Carol is a politician and she wants to make a speech to the Senate about public healthcare. But she begins the speech by talking about her childhood friend, to connect with the audience emotionally (or for any other reason). How would you describe the fact that she began her speech "from afar"?

If there is no single phrase that can be used for both Alice and Carol, two different words/phrases are also appreciated!

How about this?

"Alice broached the subject in a roundabout way."

The expression broached the subject means to "bring up" or "introduce" it. The phrase in a roundabout way means "indirectly."

Here are some ideas:

"Alice eased in to the subject."

"Alice eased in to the subject by first talking about hot air balloons."

"Alice started by talking about ants. Once Bob was warmed up to the idea, she was able to talk about spiders."

As for single words for the phenomenon you describe:

• n., circumlocution; adj., circumlocutory
• n., periphrasis; adj., periphrastic

Each of the above words can have a negative connotation, particularly when the person who is being circumlocutory or periphrastic is attempting to evade having to talk about something, or when the person does not know what he or she is talking about but feels the need to fill the silence.

For example, a person might evade having to answer the question, "What role did you play in your employer's attempt to swindle people through an elaborate Ponzi scheme?" by saying,

I'm glad you asked that question. So many people avoid asking the hard questions, so personally I commend you for the forthrightness of your question and for giving me the opportunity to respond. Moreover, I want to respond to your question, because there are probably many other people who have the same question you posed to me but were afraid to do so. I commend you, again, for your fearlessness. You've reminded me just now of an incident which occurred to me a number of years ago. Was it 1998? Or 1999? I'm not exactly sure, but the incident involved a person such as yourself who asked me a pointed question regarding a situation in which I was tangentially involved . . ..

A less pejorative way of describing the phenomenon is to use the word obliquely, as in the following:

• Alice approached obliquely the question she really wanted to ask of Bob.

Or, the word diplomatically communicates a great deal when combined with other words, particularly the word skirted. For example,

Alice wanted to ask Bob a question diplomatically, so she skirted the issue deftly until the time was right and she felt comfortable asking the question.

Again, to "skirt" an issue can involve a measure of avoidance or evasion and, consequently, some periphrasis or circumlocution, but not necessarily.

The expression "to beat around the bush" can involve empty blathering, but not necessarily. Beating around the bush can involve diplomacy, particularly when a subject or question is difficult to broach head-on. In that case, the questioner engages in a little good-natured bantering and only when the time is right will pose the question.

ADDENDUM, in response to revision of question

The technique utilized by Carol is a pretty basic rhetorical strategy which goes back to ancient Greece, where public speakers would establish their ethos in various ways throughout the speech. Aristotle suggested that ethos (modern day, "credibility") has its source in a speaker's perceived intelligence, virtue, and goodwill. Sounds to me as though Carol is establishing her credibility by enhancing the goodwill between her and her audience by reminiscing about her childhood friend.

Audience members can obviously identify with her reminiscences and may even be moved emotionally, which is evoked by Carol's use of pathos (Gk., for emotional appeals).

A few additional ideas, depending on exactly how you want to phrase it:

• "After chatting about their holiday weekends, Alice finally came around to her real subject, Bob's poorly written report."
• "Bob updated Alice on the project status, but wished she would stop dancing around the subject of the looming company merger, and what it might mean for their department."
• "After some preliminary discussion, Alice finally got to the point of her visit, Bob's upcoming sales conference."
• "Alice was always careful to tiptoe around the subject of Bob's work ethic, he was the bosses' son after all."
• "Alice wanted to ease Bob into the idea of making the presentation to the board, she knew he was uncomfortable doing public speaking."