Let's start with the basic rule on pursuits. The law enforcement officers of one country have no authority to even enter another country, let alone pursue someone (which often involves very dangerous driving), let alone use force in that other country, without the permission of the other country. This permission can be ad hoc, but countries can also enter into an agreement setting out different rules. Absent permission, the most you can do is notify foreign authorities and hope they pick up the chase when the suspect crosses the border.
There's no EU-wide rule (or at least wasn't as of 2008). The broadest relevant grouping is Schengen (among other things, getting rid of border checkpoints removes one obstacle to criminals fleeing over a border). The Schengen Convention contains a provision on hot pursuit. This doesn't let police arrest someone and take them back over the border, though; the most the chasing cops can do is detain someone until local police arrive to take them into custody pending the start of a formal extradition request (and countries can decide to not even let foreign police do that). If no request is forthcoming and the person isn't a national of the country where they're arrested, they have to be released within 6 hours.
Schengen's rules only allow pursuits for escaped prisoners and people caught in the act of committing serious crimes (countries can decide if that means a list of crimes in the convention, or just "extraditable offenses"). They also let countries set standing limits on length or time of foreign pursuits in their territory, and let countries order foreign police to immediately stop any pursuit. The pursuit can only happen on land; sea and sky are excluded (so a foreign police helicopter can't continue over the border).
Schengen countries can give each other more powers in a cross-border pursuit if they want to, either on an ad hoc basis or by treaty. There have been treaties before Schengen, and treaties since Schengen. Benelux countries have a fairly close agreement, and in a cross-border pursuit there the pursuing officer can actually arrest the suspect and take them to the competent authorities (as opposed to detaining them until local cops arrive). A pursuit under Benelux need not be of someone caught in the act, and since the intra-Benelux extradition threshold is lower, more crimes qualify for cross-border pursuit.
Source: Daman, Maarten. "Cross-border Hot Pursuit in the EU." European Journal Of Crime, Criminal Law & Criminal Justice 16, no. 2 (May 2008): 171-207.