Erasmus, the world's largest student exchange scheme, is celebrating its 30th birthday.

With more than 3 million participants since 1987, it is one of the best known and most successful policies of the European Union.

Now including adult learners, vocational students and those on work placements, in addition to university students, it has created an “Erasmus generation”, having been responsible for more than a million babies born from couples who met as part of the scheme.

About 16,000 UK students now spend a semester or a year abroad as part of Erasmus every year. France, Spain and Germany remain the most popular destinations for these students, reflecting the traditional emphasis on students taking modern language or combination degrees. However, many universities across continental Europe now offer modules in English, which has helped to increase the number of UK students able to participate who do not have prior language skills. As students strive to add distinctiveness to their CVs, the number of UK participants has increased.

In addition, the UK is one of the most popular destinations for European students, with these study placements becoming part of Britain’s cultural and educational soft power by creating thousands of de facto UK alumni across Europe.

However, while the House of Commons Education Committee believes that “continued membership of Erasmus+ would be the best outcome for the UK”, its future participation was not mentioned in the government’s recent White Paper. The government has committed itself only to considering future participation.

The question therefore is “can the UK continue to be part of it post-Brexit?”

Since Erasmus is a programme of the EU and established by EU law, the initial answer is “no”.

However, as with everything else, all depends on the exit agreement between the UK and the EU before the UK leaves in March 2019. It may be that the UK continues to be a part of the scheme up to the end of the current programme (2014-20), with future involvement subject to a separate agreement.

Other European, non-EU states are part of the Erasmus programme (Iceland, former Yugoslav Republic of Macedonia, Liechtenstein, Norway, Turkey) as permitted under Article 24 of the Erasmus Regulation. This could, therefore, be an option in the replacement for the Erasmus programme after 2020.

However, the UK will need to contribute to the financing of the programme and obey all the relevant rules without having the opportunity to influence them.

The case of Switzerland is instructive: Switzerland is eligible to participate in Erasmus as a programme country. But following a referendum in 2014 that called into question the free movement of people in its agreement with the EU, its Erasmus participation could no longer continue.

Future UK participation might therefore depend on whether it commits to free movement of researchers and students in some form. With the UK government appearing to stick to the end of free movement as a “red line” in negotiations, this does not bode well.

With only very detached participation, or no formal agreement with the EU on Erasmus, of course universities would be able to form bilateral agreements with European universities. This is, after all, how UK universities exchange students with institutions elsewhere in the world.

However, such agreements operate without funding to students, who must cover their own costs. This is likely to prevent lower-income students from participating and be extremely destructive for students on language degree programmes – which are essential to maintain given the future economic need for graduates with advanced language skills.

The argument that the UK contribution to the EU budget which covers Erasmus can be simply diverted to a national scheme does not take into account the costs of setting up a parallel system, as the Swiss example shows.

The UK government will – at the very least – need to substitute the funding awarded for UK Erasmus participation (€113 million in 2015) to enable students to participate on the same basis. No such assurance has yet been made.

For universities, the administration costs of individually negotiated grants are much higher since under Erasmus there is a commonly agreed contractual framework.

And, with less demand from EU students to come to the UK, it may be more difficult to produce viable exchanges. UK students and universities can only lose out from this.

The government’s view of a new “Global Britain” must surely include equipping graduates and young people with the cultural ability to understand the world beyond national borders of the UK and plug the post-referendum skills gap.

The evidence shows that one of the most cost effective ways to do so is via exchange programmes.

It would not only be a shame but also damaging and costly to lose the years of cooperation with the UK’s nearest neighbours and start over.

Paul James Cardwell is a professor of law at the University of Strathclyde, in Glasgow, who has published widely on EU law and politics.