The answer to this question depends entirely on how the matter is handled.

The causes of their massive arrivals are not the subject of this article, eventhough they need to be understood. The matter coinvoltes many countries and needs to be separately analysed.

Europe is now home to millions of immigrants and more are coming. They can be, a financial burden on the hosting state and local authority, a socially disruptive force, a labour disruptive force, an added component to criminality, etc.

Their problem is described as their lack of integration, but is it the only one? Can they overcome the language barrier? Can they understand and respect the different norms applied in the host country? Can the host country society live peacefully with them? What if they do integrate? Does that solve the problem?

Are there good illegal immigrants and bad ones? I dare to say yes and it all depends on the arriving immigrant state of mind and his/her aim in life.
No would be immigrant would risk such a financially costly and dangerous endeavour without a compelling reason. (war, discrimination, persecution, poverty, etc.)
Surely, it is the prospect of a safe environment, a chance for a better economic status that gives them hope. In short, a better life than what they have in their home country.

From my point of view, immigrants should be categorized and measures applied to their hosting should differ. One size fits all hosting system is the major mistake countries are committing.
War refugee is not the same as an economical migrant or the close to home immigrant.

War immigrants should be classified as refugees and allowed temporary stay, given financial support and schooling, while actively working for their safe return.
The main restriction during their stay should be the access to areas outside their encampments area so as not the disrupt the hosting communities and also to apply limits on their birth rates which from experience can be surprisingly high.
New births add the burden of additional medical care, money, accommodation space, food and classrooms.
War immigrants main aim is safety and preferably in a host country that has resources to take care of them.
However, the above is not sustainable if it becomes clear that their return is not foreseeable, the host country should with other countries sharing the same burden and strive to return those refugees back home.

Economical immigrants coming from far places such as the Indian subcontinent and south of the African Sahara are not refugees. They are driven mainly by poverty caused by grossly mismanaged economies that sent the middle class (Doctors, engineers, accountants etc.) to greener pastures, decimated small businesses through lack of credit, administrative bottlenecks, bad infrastructure (no regular electricity power, bad roads etc.) and very low education standards not able to create competent workforce at any level.
For these immigrants, effort should be made on their arrival to evaluate their education level, assess their skills and competence, try to get their criminal record if any, their age, assess their personal status (married, children etc.), get their nationality and birth place etc.
They should be considered as raw assets and taught skills the host country needs, and deployed accordingly.
If foresight is applied, they should be able to acquire and apply their skills in the host country and would probably return to their home country confidently armed with those skills and experience, maybe through employment for the host country companies overseas operations or contracts, or hopefully with money earned through their work to invest in their country or by the host country incentive encouraging return. Homesickness is a very powerful feeling, and many immigrants may likely choose to retire in their hometown, if they can afford it.

Skilled persons would likely return home, provided conditions in the home country are reasonably good, where peace reigns and the underlying causes that forced them to immigrate resolved or being actively under resolution.

Nearby immigrants are the ones arriving from the southern shores of the mediterranean, from countries with centuries of ties to Europe mostly hostile ones.(Wars and colonization)
The young ones attitude is generally disrespectful of European culture and the fact that large established communities exist already, they feel protected by their kin and engage in all sort of activities not necessarily in strict adherence to the laws of their host country, to say the least, and bearing in mind that some of them are now 3rd generation nationals.
They present the host country with a catch 22 situation, where deporting them is not really an option (many are nationals), confronting them leads to unintended consequences (religious persecution) and accommodating them creates discontent and inflames sentiments in the host society, especially as their home countries do not reciprocate in granting the same.

The first step to a solution, is to deny any illegal immigrants nationality including their children whether born before or after arrival. On the other hand, permanent residences can be given as well as encouragement to be productive and stay off social security. (Denying nationality to illegals should not affect the legal ones.)
The policy of non-interference in another country internal affairs is now obsolete, as illegal immigration is actually interfering in the host country internal affairs. Think of a country that turns them away, or organize mass deportation. The whole world will condemn it and the country in question would be forced to keep them. Is that an interference in a country’s internal affairs?

Another step is for a country or a group of countries is to create an avenue for dialog and engage the concerned source countries, with the aim of stopping or slowing departures and setting repatriation modalities, using the stick and carrot policy.
The carrot as always is financial, and the stick is a suitable and enforced economic policy that improves life and gives hope to people as migration is caused mainly by desperation.
The prescribed policy should be closely monitored and controlled. It should not be left for the home government to implement it exclusively, knowing that whatever economic policy they had in the past, fuelled the migration.

Today a similar line of thought is shaping up, a lot of thinking is leaning towards providing help to the countries that are either a staging or crossing point or the source of immigration.
However, help resources are insufficient to benefit those countries (they are many) and they will most likely miss their objective (stop immigration).
A better approach would be to select one of the large source country, agree on the road map for growth, pump it with enough funds to stabilize its currency, promote investments preferably in manufacturing and agriculture and provide investors with adequate insurance cover against any political meltdown.

That scenario should be a win-win for everybody. The source country economy should improve and can become a destination for would be a nearby immigrant instead of the costly, long and treacherous journey to Europe.
Moreover, if the right investments are made in manufacturing, the host country can become an alternative source of goods for the home country / countries.
Also, if the right investments are made in agriculture and land management, world food supply will definitely increase to the benefit of everyone.

A collateral outcome, would be from other migrants source countries, whom may see benefit from such a scheme, and request for same.

Europe was busy on itself in the past 50 years and ignored illegal immigration for too long. The numbers were not as high in the beginning and European government felt sorry and believed that it was a manageable problem.

Some European governments have not acknowledged the existence of an illegal immigrants problem. Others have and are willing to tackle it (UK extradition to Rwanda, regardless where you came from), but I fail to see it as a lasting solution, surely next government will probably suspend it as a minimum. The closest to a long term solution, is the one proposed by the Italian government, to help the source countries.
I am not familiar with the details or the timeline, but thinking along that line is the key to a lasting solution.