Walking isn't the punishment. Missing playtime at recess is the punishment. I much prefer walking to having them "stand on the X" or "sit on the red bench" like we did when I was in elementary school. The teachers know that the kids still need the physical activity and time outside to help them through the rest of the day.
Is the lack of warning strict? Yes. Is it wrong? No. I think it's better that kids learn there are consequences for our actions at a young age, when the consequence ultimately isn't that big of a deal. And if the teacher told the students that if they forget homework, then they have to Eagle Walk, then they were duly warned. I think it's great that the teacher has high expectations of the kids.
Let's teach our kids to accept responsibility for their actions and earning positive results through hard work, instead of teaching them entitlement. In the real world, you don't get a warning for missing a report deadline - you get fined, or you miss getting the account, or you miss qualifying for a grant, and you do not get the opportunity to make it up. If missing recess and taking an Eagle Walk will get my kids' attention and help them remember to turn in assignments, then I support it.