We were once a United People

The government of the USA was created to suit the people involved, in 1788 or so, and would not necessarily be suitable for a different mix of people. So what did they have in mind?

The Federalist Papers : No. 2:

It has often given me pleasure to observe that independent America was not composed of detached and distant territories, but that one connected, fertile, widespreading country was the portion of our western sons of liberty. Providence has in a particular manner blessed it with a variety of soils and productions, and watered it with innumerable streams, for the delight and accommodation of its inhabitants. A succession of navigable waters forms a kind of chain round its borders, as if to bind it together; while the most noble rivers in the world, running at convenient distances, present them with highways for the easy communication of friendly aids, and the mutual transportation and exchange of their various commodities.

With equal pleasure I have as often taken notice that Providence has been pleased to give this one connected country to one united people -- a people descended from the same ancestors, speaking the same language, professing the same religion, attached to the same principles of government, very similar in their manners and customs, and who, by their joint counsels, arms, and efforts, fighting side by side throughout a long and bloody war, have nobly established general liberty and independence.

This country and this people seem to have been made for each other, and it appears as if it was the design of Providence, that an inheritance so proper and convenient for a band of brethren, united to each other by the strongest ties, should never be split into a number of unsocial, jealous, and alien sovereignties.

And we have had a Constitution that is also made for the country and the people.

Not any more. Rapid immigration and other changes mean that John Jay's description is no longer accurate, and the Constitution will not work so well anymore.

Update: I see some Libertarians claiming that the Founders were pro-immigration, but admit:

Despite his defense of open immigration on many occasions, Thomas Jefferson wrote, in his 1782 Notes on Virginia, that America had reason to fear immigrants from "absolute monarchies," because "[t]hey will bring with them the principles of the governments they leave, imbibed in their early youth; or, if able to throw them off, it will be in exchange for an unbounded licentiousness, passing, as is usual, from one extreme to another."

He also cites this Cato Inst. article:

Alexander Hamilton argued that the “advantage of encouraging foreigners was obvious and admitted,” asserting that “persons in Europe of moderate fortunes will be fond of coming here where they will be on a level with the first Citizens.”

Yes, America was under-populated, and could obviously from Europeans of moderate fortunes who can immigrate and be on a level with Americans.