His warnings come amid the continuing row over the future of the border between Northern Ireland and the Republic.
The British government wants to keep the soft crossing by imposing high-tech surveillance which would create a "virtual" border.
The Garda equality officer told the Guardian: "This is the first time uniformed Garda officers including myself are marching in a Pride parade anywhere on the island of Ireland".
"The imposition on Brexit, against the will of the people, is a serious challenge to our economy, our agreements, and to the rights of citizens". With leaving the European Union remaining opposed by the majority of Scottish people, a botched, or failed Brexit, could easily cause the question of independence to rear its head again. And, to quote Michel Barnier, the clock is ticking.
"I totally appreciate that this it is an issue for the Northern Ireland Assembly, but we need an assembly up and running".
The 27 countries that are still in the European Union are due to hold talks in October to decide whether or not enough progress has been made on the divorce settlement to allow trade talks to begin with the UK.
The Joint Committee on the Implementation of the Good Friday Agreement published the report, which was compiled by Senator Mark Daly on Wednesday.
"Students have led many changes over the past and we should not be underestimated".
He is then expected to meet with local political parties.
Saturday's breakfast, in the The Northern Whig pub in central Belfast, was also attended by Michelle O'Neill, the regional leader of Northern Ireland's largest nationalist party Sinn Fein, and by a number of gay rights activists from the Irish and Northern Irish police forces.
He said that he also favoured the United Kingdom joining the European Free Trade Association (EFTA) - alongside Iceland, Liechtenstein, Norway and Switzerland - as a way for the country to remain associated with the EU single market.
Mr Mulhall also said that Ireland hopes Britain will remain in the EU customs union post-Brexit.
As for the taoiseach and Theresa May getting directly involved in the Stormont talks, that would be a welcome step which would put some much-needed impetus into the stalled process.
Varadkar described Brexit as "the challenge of our generation", saying, "Every single aspect of life in Northern Ireland could be affected by the outcome".
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