What is Debian's equivalent of do-release-upgrade to upgrade the operating system? (for example, from stretch to buster)

Debian does not provide a single command to upgrade the OS to a new release. The Release Notes for each release include upgrade instructions for supported hardware architectures.

You can find release notes for all Debian releases via the Debian Releases page.

For example, to upgrade a 64-bit PC from stretch to buster, follow the instructions in Chapter 4. Upgrades from Debian 9 (stretch) under Debian 10 -- Release Notes for Debian 10 (buster), 64-bit PC.

You should always be able to find the release notes for the current stable release at https://www.debian.org/releases/stable/releasenotes.

Although upgrading a Debian release from "oldstable" to stable is usually painless, it's important to follow the Release Notes because the OS can differ from release to release in ways that could affect your specific installation.

The Release Notes also contain information and tips about changes in the new release that can save considerable time and effort. For example, the upgrade process for some previous releases recommended the use of aptitude for the upgrade. For upgrades from stretch to buster, the apt tool is recommended instead of aptitude. (Although aptitude is suggested for resolution of some problems after the upgrade.)