For an API product, how have you been able to successfully continue engagement with developers after they have completed onboarding? Strictly email? Does that outperform/under perform regular web and mobile customers?

Newsletters are great--to a developer or not, email marketing has a ton of value. At my last company, email was the #1 driver of actions–that was consistent with web and mobile customers as well. But there are a handful of other mediums you can lean on as well. YouTube, Stack Overflow, Twitch, Reddit, Twitter, and LinkedIn all have done pretty well. Also at my last company, someone on my team had the idea to run Google Display ads, and they performed really well. Depending on the size of your organization you may also have a Developer Relations team. Hosting events or webinars/livestreams also tend to perform well. And don't forget your website and your blog, because your developer content is often one of the first places a developer will go to find out information. So think of it like marketing to any customer base: consider where they are in the funnel, think about what actions they need to take to get to the next step, and then leverage the mediums you have at your disposal to push them through the funnel. 

I like the spirit of this question, as it's not just relevant to API products but also any product that has a similar onramp due to it being technical. You also touch on something that many inadvertently forget--that it's not enough to launch a product, you also have to think about the "landing" and how to drive continuous engagement.

Here are the few things I've seen teams do: 

• At the product level, you want to monitor API usage, and depending on the behaviors you're trying to drive, figure out whether they're hitting the points of interest that don't just denote that they're onboarded (assuming you have some kind of tutorial or "hello world" moment in mind), but that they've made a bunch of calls, connected the API to an app or workflow and moved whatever process they're trying to build into production. Typical PLG phases track to this i.e. activation > adoption > standardization > advocacy
• For marketing, I can share what we did when I led developer marketing at Mulesoft as an example. There we ended up building three specific user journeys that tracked a developers progress across the process of consideration (mostly marketing driven) to onboarding and adoption (product and customer success driven) and used a mix of tactics to lead them through it: email nurtures specific to the intent we had flagged at the time of sign-up (e.g API design and management) to get them situated, interactive html5 walkthroughs when they first logged into our platform based on that intent (implemented WalkMe for this), and then based on usage patterns would trigger follow-ups from CS on how they were doing and whether they needed help (that latter part was a bit higher-touch but lucklily this was a function I could rely on given how important onboarding and going from PoC to production was important to what we did). 
• Finally, I think it's really important to highlight the successes of other developers on your platform and make sure that devs who are earlier in their experience with your solutions can see this. You want to make it aspirational. One of the things we did really well at Box for example in the early days (or even Salesforce for that matter) was tout the successes of our developers and partners across our channels. For example, featuring a developer/partner spotlights on your blog, doing a homepage takeover or carousel item that highlights a specific developer project or partner, and even leading with external signage/billboard that puts those developers first are all things that demonstrate that your organization lives and breathes their success. If you're a pureplay platform company, you're likely already thinking about this, but if platform is an adjacent GTM (e.g Salesforce, Slack, Okta, Apple), you need to make sure that marcomm understands this and keeps it in rotation alongside your company's other narratives. 

In product is good, but only if it is helpful. Do not make your product times square. Release. notes, blogs, social leading to blogs, whats new section in product, office hours, newsletters, etc. work well. Just give them the option to self serve the information they need and stay out of their way for a solid product led growth funnel for developers.