Authority gets compliance. Influence gets people who actually want to help you, and the second one tends to outlast the first, especially once you leave the room.
Most people's first experience with influence-without-authority is a project that spans multiple teams, where you need something from people who don't report to you and have no obligation to prioritize your request over their own manager's. Trying to force that with urgency or escalation usually backfires, you get compliance now and reluctance every time after. The people who consistently get things done across teams they don't manage are working from a different set of tools entirely.
The first tool is simply making the ask easy to say yes to. Vague requests like "can you help with this when you get a chance" invite indefinite delay, because there's no clear scope and no clear deadline. A specific ask, what exactly you need, how long it'll take, and by when, gives someone a fast, low-friction way to say yes, and most people will if the ask is genuinely reasonable and clearly bounded.
Instead of: "Whenever you get a chance, could you take a look at the API docs for this?", try: "Could you review the API docs section I flagged? Should take about fifteen minutes, and I need it by Thursday so I can finish the integration on time."
The second tool is understanding what the other person actually needs, and framing your ask in terms of that, not just your own deadline. A team that's being asked to help you is far more likely to prioritize it if they can see what's in it for their own goals, fewer support tickets later, a dependency cleared off their own roadmap, credit in front of their own leadership. This isn't manipulation; it's simply recognizing that everyone is juggling their own priorities, and your request is competing with all of them.
The third, and probably most durable, tool is reputation carried over time. If you've been the kind of person who follows through, gives credit generously, and doesn't waste people's time with vague or last-minute asks, you'll find people say yes to you faster than they say yes to someone with a more senior title but a worse track record. Influence compounds. Every reasonable ask, delivered well and followed up on properly, makes the next one easier.
The one thing influence can't substitute for is a genuine conflict of priorities. If what you're asking for is a bigger commitment than the other person's own manager would sign off on, no amount of good framing changes that, at that point, the conversation needs to happen at the manager level, not through personal goodwill alone. Knowing that distinction is itself part of using influence well.