I think small changes are more effective than big changes and end up being more sustainable for a longer period of time. Therefore, I'm proposing small changes within existing activities to help integrate empathetic thinking on a daily basis.
News show: Many schools have a news show, especially at the elementary school level. Schools could incorporate empathy challenges on a weekly basis, then give shout outs the following week to students seen exhibiting the targeted skill/behavior. Challenges could be differentiated for lower and upper elementary students. This could also tie in with a school-wide PBIS.
Compassionate Conversations: Students are randomly paired up with each other at lunch and are tasked with making meaningful conversations. Conversation topics could be provided to help guide the students' conversations. The randomization of seating would help encourage students to speak to others who aren't in their close friend group. Once a month, grade levels could mix to provide empathy across ages. This could help students learn meaningful things about not only their classmates, but others in the school to know peers on a deeper level.