How to go about influencing people across different functions and teams?

Influencing people across different functions and teams is very similar to what I outlined under the question "How do you manage peopel who don't necessarily report to you?" This includes things like:

• Get an understanding of the team and what they care about
• Involve them early and often 
• Be respectful of their time
• Provide context on where you're coming from 

Few tips here.

First, build trust before you need it. It’s so important to build these relationships before you need to test them. Grab coffee, lunch, ask about people’s family life, etc. If you’re able to build a foundation of trust before you need to influence a different team, you’ll find it’s much easier than starting that relationship cold.

Second, understand their priorities. It sounds obvious but you’ve got to start by putting yourself in their shoes. There are some easy ways to do this. As an example, you can start your conversations by asking what the other person or team’s upcoming goals or priorities are. That’s a good signal to them that you’re not going to come in and push your priorities without understanding theirs first. The other good way to do this is to be great at steel-manning (mentioned in another answer). When you can articulate another team or function’s perspective as good as or better than that team can, you’ll gain their trust.

Third, be helpful in unexpected ways. Once you understand the priorities of other teams, you can be proactive in helping with what they need. If you understand their priorities in advance, then you can find small ways to help them, which builds the relationship and engenders trust. I find that even small gestures like connecting to experts in their area, or giving thoughtful feedback goes a long way in building trust that helps with influencing a team later.

It's often overlooked, but I believe it all starts with being genuinely interested in other people's perspectives.

1. The more diversity you can include in your thought process the better the output will be. And in a culture where the "best idea wins," this sets you up for success because there is a lot of research that shows the best ideas start with the most diverse inputs.
2. People want to feel like their opinions matter and that they are being heard. Being genuinely interested is a way to demonstrate that you care about what people think. This doesn't mean you are always going to do what they want but, they were part of the conversation and consideration. Even if they see a sliver of their perspectives reflected in the position you are trying to drive, it shows that you did listen.
3. Influence is a give-and-take game. People often mistake "being influential" with "never being swayed but being genuinely interested in the perspectives of others means you just might have your own opinion swayed. Showing that you can adopt positions you didn't originally hold deposits "influence capital" into your relationships that will be there for you to withdraw someday.
4. Finding flaws in your own thinking. Face it, you aren't perfect nad you don't know everything. A way to lose credibility (and influence) is to come off as materially wrong. Not just you forgot a fact, but rather you are not in command of ALL the facts. The perspectives people have around you can fill in those gaps before you make a material mistake.
5. Understanding motivations. Genuine interest in people is also a way to understand their motivations, and understanding motivations is a fast path to being influential. People do stuff that benefits them - this is not a bad thing it's just a fact. If you understand what motivates a detractor, consider how you can reframe or adapt your point of view to better connect with their motivations. To be clear I'm not talking about quid pro quo and negotiation here. I'm talking about finding ways to demonstrate to your peers and partners that you understand them so that they see how your point of view helps them achieve their goals.

The best way to do this is to start with building a good relationship. Get to know them as a human, show interest, and share who you are as a person as well. Be proactive and schedule time - use it to ask for their opinion on your work and get feedback - people love sharing their perspective and being asked for feedback and it's a great way to start building trust. Understand their goals and find ways to make their job or life easier. Getting these wins will show your competence and care and are a great start to their openness to listening to you and your ideas. As you get to know them understand what makes them tick and then if you are trying to get their agreement on something follow two simple rules 1) Frame it in a way that aligns with something that is important to them (not, we need to remove this line from our checkout, but we want to improve conversion and sales by X%, if we remove this field we think we can) 2) Adjust your communication style to their personality or preferences - for some it may be data, for others it may be seeing something cool, for others it might be having confidence that we are preventing something very bad from occurring