We live in London, their mother is from Northern Ireland. As you know many Britons apply for Irish passports since the referendum results.
Do I have to drag the children all the way to Ireland in order to apply for their first Irish passports?
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Thankfully, it appears you don't have to drag them to Ireland to apply ;)
The Embassy of Ireland Passport Office London, is located at:
114a Cromwell Road, London, SW7 4ES
The Passport Office is open to the public from 09.30 - 16.30 Monday to Friday.
If you choose to post your application, we strongly recommend that you use registered mail or a reputable courier service.
Given that last paragraph there, you don't even need to drag them to the Passport office in London.
The page I linked to above has full details about what your application will require. In your situation, assuming their mother was born in Ireland, it looks like you'll just need the children's "original long-form birth certificate[s]" and the "civil marriage certificate (if applicable) of your [children's] Irish-born parent".