Not all features available in the server version are available in the cloud version of the app and visa versa. See the table below for differences.
Release Reports build column and warnings
None of these differences have an impact on the ability to migrate data, these differences only effect how the app shows data within Jira.
Simply install the Jira Cloud Migration Assistant and the upgrade the Jenkins Integration for Jira app to the latest version available, and migrate all your Jenkins data within Jira to your Jira Cloud instance.
Both the https://marketplace.atlassian.com/apps/1221134/jenkins-integration-for-jira-lite?hosting=server&tab=overview and the https://marketplace.atlassian.com/apps/1211376/jenkins-integration-for-jira?hosting=server&tab=overview Jenkins Integration for Jira - Lite and the Jenkins Integration for Jira support automated migration paths using the Jira Cloud Migration Assistant.