About the Trinity-of-the-Mountains concordats (1828, 1974, 1999, 2005)
There are five concordats in this strange series: the first made by a bloodthirsty Bourbon in 1828, and subsequent ones in 1974, 1999, 2005 and 2016. The follow-up pacts look like foot-in-the-door concordats. Although the Vatican website correctly identifies these as “Concordats”, they are officially titled “Endorsements” (Avenants), which hides their function from the French public.
A "secular" mission for the Sisters of the Sacred Heart of Jesus?
Sister Josefa Menendez (1890-1923) is the best-known member of the Society of the Sacred Heart of Jesus. She died young, less than four years after entering their convent, but during that short time maintained a strenuous pace as messenger of God, receiving revelations "quasi daily".  (Those wishing to avail themselves of her demonstrated access, can find a prayer in French asking her to intercede on their behalf.) 
Today the Sisters of the Sacred Heart of Jesus are a large multinational with branches in 45 countries; in 2002 they gained observer status at the UN.  Curiously, this huge organisation was not in a position to support its convent in Rome — either to finance renovations to the building or to recruit successors for its aging nuns.
The concordat series stretches from the Bourbon Restoration until the present, with pacts under various names dating from 14 May and 28 September 1828, 4 May 1974, 21 January 1999, 12 July 2005 and 25 July 2016.
The 1999 concordat below was still in the name of the Sisters of the Sacred Heart of Jesus, but it prepared the way for a takeover. In its first article it stipulates that the aging nuns can "call upon" other orders.  A few years later the French Government seems to have begun wondering what was going on.
In November 2002, the French Embassy to the Holy See, which had obtained considerable funds from the French government to renovate parts of the Trinità, approached Sr. Clare Pratt as Superior General of the Society of the Sacred Heart, to ask what our plans were for the future and, in view of undertaking more renovations, to demand a significant reinforcement of personnel for the coming years. 
The answer came in the next concordat on 12 July 2005. After the French Government had renovated the convent, it was claimed to really be the property of the Minims (Monastic Communities of Jerusalem) which has no monasteries in France. The concordat asserted that, due to the "impossibility" of the aged nuns carrying on, the property was to be transferred "back" to the Minims. 
[The Vatican stresses that Trinity-of the Mountains] was never owned by the Society of the Sacred Heart. It had belonged to the Order of Minims for over 300 years and was confided to the Society by the Diplomatic Conventions of 14 May and 8 September 1828 between the Holy See and the Government of France, for the purpose of educating young girls.
Although the new owners, the Minims, are absent from France, the Trinity-of the Mountains property was still considered French in view of the fact that it belongs "to the 'Pious Establishments of France at Rome and at Loretto', a French foundation established in Rome for centuries, whose possessions consist of five churches and more than ten properties in the Italian capital".  However, the very next year, in 2006, the Trinity-of the Mountains was transferred once more, this time to the Monastic Fraternities of Jerusalem, then ten years later, to the Emmanuel Community. 
Leaving aside this impenetrable thicket of property dealings, two things are clear:
First, the concordats of 1974, 1999, 2005 and 2016 are all successors to the concordat of 1828 made during the reign of the restored Bourbon King, Charles X. In the few years before he was toppled by a popular uprising, Charles X dissolved the Chamber of Deputies, censored the press, restricted suffrage, brought in a law imposing the death penalty for sacrilege — and concluded this concordat.
And second, the purpose of the original concordat — the education of girls by a teaching order — no longer applies, for in 2006 the nuns, who were dying out, handed it over to the Monastic Fraternities of Jerusalem. Ten years later they passed it on to a French charismatic order, the Emmanuel Community. Yet the Vatican has managed to find a justification for retaining the "Endorsement". After all, this concordat sets a precedent for further privileges for the Vatican in that most secular of countries, France. The Church is not about to give up this valuable wedge.
The French Government gives an account of the "History of the church and convent of the Trinity-of-the-Mountain" (L’histoire de l’église et du couvent de la Trinité-des-Monts).
1. "Vie de Josefa Menèndez", Message du du Sacré Cœur de Jésus à Soeur Josefa Menèndez, 11 mars 2007. http://www.oeuvre-du-sacre-coeur.be/spip.php?article13
2. "Prière pour obtenir des grâces par l'intercession de Soeur Josefa", [Imprimatur: Montepessulano, die 1 fabruarii 1848. Jean Rouquette, Vic. Gén.] http://prieresetdevotions.blogspot.com/2007/02/soeur-josef-menendez-poitiers.html
3. Religieuses du Sacré Cœur Jésus (RSCJ), "Frequently asked Questions about our Presence at the UN", http://www.rscjinternational.org/content/view/2118/227/lang,e/
4. Religieuses du Sacré Cœur Jésus (RSCJ), "The Trinità dei Monti - Rome, Italy", 13 October 2005. http://www.rscjinternational.org/content/view/148/47/lang,e/
6. Vatican Information Service, "Accord entre la France et le Saint Siège à propos de l’église de la Trinité des Monts", [no date]. http://catholique-lyon.cef.fr/article.php3?id_article=611
7. Religieuses du Sacré Cœur Jésus (RSCJ), "The Trinità dei Monti - Rome, Italy", 13 October 2005. http://www.rscjinternational.org/content/view/148/47/lang,e/
8. Vatican Information Service, "Accord entre la France et le Saint Siège à propos de l’église de la Trinité des Monts", [no date]. http://catholique-lyon.cef.fr/article.php3?id_article=611
9. "France, Holy See update agreement on French church in Rome", Catholic Culture, 25 July 2016. http://www.catholicculture.org/news/headlines/index.cfm?storyid=28912
“Avenant” tra la Santa Sede e la Repubblica francese alle Convenzioni diplomatiche e agli “Avenants” relativi alla chiesa e al convento della Trinità dei Monti, 25.07.2016