The person holding the Irish passport is an EU citizen and has all the rights and obligations that British citizens had before they decided to shoot themselves in the foot. EU citizens have the right to move freely within all the EU/Schengen countries/states and have the right to stay and work in...
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... whichever of these countries/states that they choose to. If they choose to relocate permanently, then they should follow the ruling for the country/state that they want to move to. For example, before Brexit, UK citizens could move freely and work in Spain but they were obliged to follow the rules for applying for residencia after 90 days, which most people did. Those who never followed the rules are those who are no longer able to stay in Spain for more than 90 days in every 180. The same rules still apply to Irish, French, German and other EU country/state citizens. You have the right as an EU citizen to move to Spain but need to do this under the current rules regarding freedom of movement. Irish passport holders would need to prove that they either have a job to go to or have a pension to live off or have enough money in the bank to be able to support themselves. They don't need to show that they have a huge amount of money in the bank nor apply for a visa nor a TIE, which is for third-country citizens only.