The phrase 'fall under the responsibilities' is correct and usable in written English.
It is a phrase used to refer to something that is included in a particular set of responsibilities. For example: "Teaching and grading tests fall under the responsibilities of the school's teachers."
Previously these closely linked areas were split between two different ministers, now they both fall under the responsibility of one fully fledged minister of state.
All the analysis, demonstration, and examination work commonly does not fall under the responsibility of the people from any single functional organization.
Particular focus is directed towards the three volcanoes that fall under the responsibility of the Institut de Physique du Globe of Paris, namely La Montagne Pelée in Martinique, La Soufrière in Guadeloupe, and the Soufrière Hills in Montserrat.
Due to the risks associated with MEC identification and removal, MEC removal and subsequent land management fall under the responsibility of multiple government agencies in concert with private companies.
The UK government is currently in talks with Scottish ministers on devolving powers which it says are needed to make the referendum legally watertight, as constitutional issues fall under the responsibility of Westminster.
Vaccination facilities and the organizing of outreach and fixed strategy sessions fall under the responsibility of the departmental authorities.
The situation is even more complex because not all environmental problems currently fall under the responsibility of the districts.
One should be aware that many of the activities carried out by community support providers should ideally fall under the responsibility of the public service.
According to the law, its approval therefore did not fall under the responsibility of a committee for the protection of persons (CPP).
It might be that nurses give more weight to aspects of their work over which they can have control than to those that fall under the responsibility of the organization.
