I am not aware of a way of bulk changing issue links. Although there is an option that is similar. It is possible to link multiple issues to another issue at the same time. At the "normal" link window there is the possibility to search for issues.
There you get an option to make an advanced search using JQL. After searching with JQL you can select multiple issues to link.
Then press "Add", and now you can link them all to the original issue.
There is a limitation to this solution. You can only link 10 issues at a time, so a big bulk-operation can not be substituted.
A workaround to the limitation is to run a filter from issue navigator, export all issue keys to Excel, paste the issue keys in a note, exchange the newlines with spaces or tabs and then paste it into the issue field on the link-view.
Hope this helps and good luck with your bulk-problem!
For recent Jira (writing this on 2021) Linked Issues a bit different (at least in our case we need to go through no. 1):
See below images for references:
Hi everyone, We’re always looking at how to improve Confluence and customer feedback plays an important role in making sure we're investing in the areas that will bring the most value to the most c...
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