He was found in a holdall, behind a roll of old carpet in a communal hallway, with a Tesco bag over his head.

Paulius Dzingus had died within hours of his birth, in Fraserburgh, in April 2010. His mother, a newly arrived migrant called Ineta Dzinguviene, had put clingfilm over his face.

Just a year earlier Dzinguviene had smothered another child, Paulina, in her home town of Marijampole, Lithuania. This time she had used a polythene bag and dumped her baby's remains in a suitcase. That crime may never have been discovered if it hadn't been for increasingly successful links between police and law enforcement in Scotland and the rest of the European Union.

The 29-year-old is now serving 15 years for each murder, after being tried first in Scotland, in 2011, and then, in 2014, in her native Lithuania.

This didn't happened by accident. It was the result of a textbook international co-operation between law enforcement and prosecution in two European Union member states.

Advocate Depute David Dickson, of the Crown Office's International Co-operation Uni, was at the heart of talks, brokered through Eurojust, the EU justice co-operation organised in The Hague.

"It started with a letter of request we issued for some background about Dzinguviene," he said. "A [Lithuanian] police officer went along to the family house, knowing that the body of a child had been found in Fraserburgh.

"He came across the body of a baby in the same circumstances in the suitcase with cellophane over its face.

"That then led to a murder investigation in Lithuania, only because we had made this request for background.

"That meant a question over jurisdiction. Lithuania can prosecute crimes that have taken place abroad. We can't. There was the potential the Lithuanians could seek to assert jurisdiction over our case - over the woman we had in custody for killing a baby in Fraserburgh.

"They could, in theory, have tried her for both murders. We could only prosecute for one, not the one in Lithuania, which would have to be done over there."

How did they figure out what to do? They went to The Hague. Police and prosecutors, their trips and translators funded by EU, quickly resolved the issue at Eurojust.

Dickson explained: "In terms of the public interest, it was agreed that we would deal with our case first, because it was closer to indictment."

But the two sides also agreed to co-operate on a draft European Arrest Warrant - the document that has done more than any other measure to ensure EU nationals cannot seek refuge from justice in another member state.

So the Eurojust deal meant that, whatever the verdict, there would be a sting in the tail for Dzinguviene when she went for trial for the murder of her baby at the High Court in Livingston.

"If she was convicted, we would arrest her for extradition to Lithuania," Dickson said. "And if she was not convicted we would arrest her, too."

In fact, she was convicted. As a life prisoner, she could only be transferred out of the country to stand trial with the permission of the Justice Secretary. That permission, also arranged through Eurojust, was granted, provided she stayed in custody and was returned to serve out her Scottish sentence after trial. Which she did.

The Dzinguviene case was relatively straightforward. She committed two crimes, one in Scotland and the other in Lithuania.

But there are plenty of crimes that are truly cross-jurisdictional. And that, Dickson's colleague Malcolm McBain, reckons is when Eurojust becomes "fantastic".

The prosecutor, whose job is to get evidence rather than people across borders, is a regular visitor to The Hague. He explained why: "Let's say we have a drugs routes through France, Spain, Belgium, Holland, England and Scotland.

"There is no point in there being separate investigations.

"It's one crime, a transit crime. So who prosecutes that? Spain, where the drugs come in? Or Holland, where they were transited. Or Scotland, where the end user was? All are valid. There is not one that is stronger than the other. So how do we decided where to prosecute? Eurojust.

"It is almost like the United Nations. Each country gets two places, one police officer and one prosecutor. We sit down, each with a microphone - only one of which can be switched on at once - with a bank of translators behind - and we thrash out who is going to deal with an issue."

McBain routinely deals with requests for evidence, for witness statements or documents. But this is quicker through Eurojust, which creates joint investigation teams, funded by the EU, for sharing evidence.

Police Scotland has an officer permanently based at Europol, Eurojust's sister body for law enforcement, also based in The Hague

McBain and other international prosecutors recently had to decide how to deal with a man who sexually exploited children over the internet. This, he admits, is the ultimate global crime: victims of "sexploitation" can be anywhere; so can perpetrators.

He explained: "This was one of these cases where youngsters are contacted over the internet by a paedophile and then groomed, coerced in to providing pictures or taking their clothes off.

"As soon as the paedophiles have one picture, they get more by blackmail, saying something like 'I am going to put your picture on a porn site'.

"Well, in this case we had a handful of victims in Scotland, some more serious than others.

"We also had victims from Norway, England, the USA, Canada; from all over the world.

"The accused was found in another country. How do we deal with this? Do we want to have our own prosecution, let another country take over our prosecution? Initially every country wants to go themselves.

"There was a clear pecking order: who had the most serious victims, where the accused was actually caught.

"We were quite low down the pecking order. If we all took turns to prosecute, by the time he got to Scotland, he would have been jail for years.

"The offences in Scotland probably merited a two-to-three-year sentence."

Cue jurisdictional bargaining in The Hague. "The English were very keen to prosecute alone," McBain says. "I was keen to push the case to the authority who had the accused. Why risk having a small prosecution when we could be part of a much bigger one?"

But the Scottish prosecutor had another consideration: here in Scotland, as in England, courts operate an adversarial system.

Witnesses, the children, would be cross-examined by lawyers representing the man they had accused. Not on the continent.

"It would be a inquisitorial system, it will be done by an investigative judge and they will not be cross-examined.

"If there are any questions for the defence they will be asked through the judge."

The youngsters, therefore, would be spared the ordeal that so many complaining witnesses suffer in British courtrooms. They would be quizzed, yes. But by somebody neutral.

Officially, it is up to the Lord Advocate, Frank Mulholland, to decide whether to relinquish any claim to prosecute.

In this case - details of which cannot be revealed for legal reasons - he agreed not to pursue the case in a Scottish court, because the chances for justice were higher in another jurisdiction.

So did the English prosecutors. Did McBain talk the English out of trying to claim the case? "I don't want to say if I did or I didn't but I may have helped in that regard," he said. "I was quite adamant that the UK should be passing this on."

The Scottish victims were contacted before a decision was made. Recently, too, a foreign jurisdiction sought to prosecute a Scottish suspect in what McBain coyly refers to as "another internet thing".

First, he wants to be sure there aren't other victims here. But he adds: "We would happily entertain another country taking over the prosecution. But if there are a number of victims here then we would take over ourselves because we have the suspect in our jurisdiction."

Are there jurisdictional wars? "Yes and no," replied McBain. "There are countries which are more muscular, or less co-operative than others. There are chief prosecutors, too, who like to grandstand on the international stage. But not in Scotland. It doesn't matter who gets the man, it matters that the man is got."