To develop wordpress pages on Kubernetes, there are two types of tools you might want to use:

1. Dev Tools for Kubernetes Workflow

There are a couple of dev tools for building applications directly on top of Kubernetes:

• ksync: https://github.com/vapor-ware/ksync (basic code sync)
• DevSpace: https://github.com/covexo/devspace (real-time code sync)
• Skaffold: https://github.com/GoogleContainerTools/skaffold (continuous deployment)
• Draft: https://github.com/Azure/draft (continuous deployment)

2. Tools for Managing Dev/Staging/Prod Environments

I have used several tools to manage, merge and update dev/staging and prod environments in WordPress. Here is a list:

• DB Migrate Pro: https://deliciousbrains.com/wp-migrate-db-pro/ (basic version free, very reliable and fast)
• Versionpress: https://versionpress.net/ (free, based on git, allows branching and undo, but is horrible to use with plugins)
• WP Staging: https://wordpress.org/plugins/wp-staging/ (creates a staging site in a sub-directory, might not be optimal for container-based WP dev)