We don't really persuade people. They persuade themselves. But here's a way to influence their thinking.
'Put yourself in the other person's shoes' might be a cliched instruction, but it should be in your persuasion toolkit.
Make it a habit in your team.
Let's have an example
Let's say you want to persuade your colleagues that they could improve their customer service.
You could invite everyone to consider the question, 'What it's like being a customer of ours? (Include in your customers people from other teams - your internal customers.)
Most of us are so subjective that we would probably just think of something superficial: 'Generally okay', 'We'd be reasonably happy'.
Ask them to imagine being a client who comes into the office for the first time. 'What would your first impressions be? Would you feel welcome? Could you think we looked disorganised or unprofessional? What would give you that impression?'
Ask them to imagine being a colleague from Accounts who urgently needs a copy of an invoice. Imagine she sends you an email, then calls, but you've taken a leave day to go fishing. What would happen next? Could anyone else provide the invoice? Could anyone else contact you? How would you feel if you were her?
Why it works
Wearing another's shoes is challenging, but it disrupts our natural tendency to overestimate our success and abilities. It shakes us out of complacency and our natural tendency to see the world from our own perspective.
Keep probing and be specific. 'How would you feel, in this situation...?', 'What if...?'
I use it in training. In writing: 'How would you react if someone sent you that email?, 'How about this one?'