I actually led an effort to require every single person in our company to work for a morning or an afternoon in every other person's job. We called this effort "Walking in Each Other's Shoes". The intent was for people to gain a better understanding of what their peers were "up against".
Then we started our lean journey, did some kaizen events, and then stumbled onto using temporary, cross-functional, self-directed work teams and A3 thinking to do hundreds of improvements every year. More importantly, that meant there were thousands of opportunities for people to learn to work seamlessly together. How do you make great TEAM players? Get people to work on lots of teams!
One of my core beliefs is that if people are together a lot, "respect for people" and empathy will follow. Another core belief is that people are smart and people care. People WANT to do a great job and contribute to the prosperity of their peers. Nobody gets out of their car in the morning intent on screwing things up! The more people are together, especially working together to make improvements, the more they realize "I'm OK, YOU'RE OK". Pre-conceived notions die, trust and appreciation builds.
Figure out ways to get people from different functions together more and more. Huddles, GEMBA walks, and A3 meetings and closings are examples of ways how you can make empathy standard work.
God knows our nation could use a little more empathy right now.