Is CI/CD approaches or the best practices will differ for an enterprise application which is on-premise to Cloud based? Usually the bug fixes or patches for an enterprise application (following micro services paradigm) goes seamless but in case of on-premise which follows the Monolithic approach has to be installed on all the servers manually.
So I would say that CI/CD approaches whether on premise or in the cloud will share certain characteristics. Like for example having an automated test suite that is triggered through version control should be a common theme.
If you are using Jenkins for example you have the ability for much more fine grain customization than if you use Travis CI and Circle CI which are more often used with Open Source projects and with Bitbucket and Github for example.
Whether you use Travis CI, Circle Ci or Jenkins for example you still want to have fast builds with a good test suite composed of Unit and Integration Tests, have code linting and style checking, and a possible compilation step with application binary versioning.
One of the most important things that you want with your CI Build is that the feedback loop is fast, if a code change has caused some breakage you want your CI build to immediately notify you.
You can easily setup Jenkins to watch for code changes pushed to a remote repository. Anyways I hope this answers your question.
You can have additional builds for an Acceptance Test Suite, Load Test, perhaps a suite of security tests. All of these builds should be automated and part of a larger Deployment Pipeline and if you follow Continuous Delivery these automated build should serve as quality gates where you may not necessarily be deploying continuously but your software has gone through a series of automated builds.