If yes, how do they do it and what are the advantages compared to generic lead generation process?
Take a look at something called lead scoring. Most of the marketing automation and sales CRM systems use it to one extent or the other.
There are many ways to do this. If you're interested in a customer's "Content Journey to Conversion" within your own family of websites, you assign people session ID's when they land on any property, and fire async JS events on every page/content item they consume.
The Enterprise solutions (Adobe Marketing Cloud and Tealium come to mind) allow you to build dynamic customer journeys where each user action affects the communication. For example, the customer registers on your website, you send him a welcome letter, a few days later he browses some product category, he gets a "You may also be interested in" letter which affects the normal path where you send a follow up two months later.
Of course the golden goose is the product that monitors everything I do, measures intent, and at the moment I am ready to buy delivers me precisely targeted relevant information that is tailored to my specific need. We're not there yet but it's closer than people think.
Do you mean triggering based on activity on other sites, or based on their activity on your website. Both are possible, and the more information you have the better. Sophisticated Marketing Automation platforms with good add-ons can do both.