I am afraid that there may be no straightforward method to merge 2 JIRA instances. However in order to merge your JIRA instances, probably you can try to use the Project Import tool, whereby you can import the projects belong to the source instance to the target instance.
And if this is the case, yes you may have to upgrade your JIRA 5.1.4 to JIRA 5.1.6 first, before you can import the projects to the target JIRA 5.1.6. If you don't want to upgrade your JIRA 5.1.4, probably you can deploy a staging JIRA 5.1.6 instance for you to upgrade your data.
Anyway, there's a KB article written outlining the step to merge your JIRA instance, and you may kindly refer to it for more info on this:
- How to merge JIRA instances
I hope that this will help!
If you just want to merge the issues only, another approach you can try would be export JIRA issues to CSV: https://marketplace.atlassian.com/plugins/com.atlassian.plugins.jiraexportissuetocsv
Then import the issues to your target instance using CSV importer in JIRA: https://confluence.atlassian.com/display/JIRA/Importing+Data+from+CSV
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