Propagate parameters into Lambda

We see there is a way to inject parameters into Lambda on initialization as described in this post: Sharing Secrets with AWS Lambda Using AWS Systems Manager Parameter Store | AWS Compute Blog

It’s not clear to me then though when and how often the Lambda would be re-initialized. So when we update a value, how do we get it to propagate into our lambdas?

Ideally we would like to have something like envconsul do it which watches the values and when they change, forcibly restarts the process that it’s running. Is there a way to do something like this?