The short answer is 'Yes.' The long answer would be a beast to figure out and write considering the vagueness of the question. However, I can give a decent case study for you in the form of Liberia.

Liberia, a tiny country in West Africa, was founded by Afro-American settlers with the aid of the American Colonization Society. The ACS was actually run by white Americans, who wanted African-Americans to 'return' to Africa (I put that word in quotations because some of those sent to colonize were people with no connection to West Africa. So it wasn't much of a return) due to ideas of abolition, or African incompatibility with the west, or, really, a number of confusing reasons. What's important is that President James Monroe approved of this idea and the Society's mission, and he used the Slave Trade Act of 1819 to send the first Afro-American settlers to the area that is present-day Liberia. These settlers became the Americo-Liberians. That's a brief history.

In 1847, Liberia declared independence, and its constitution was deliberately modeled after the Presidential System of the USA. Which made sense, considering its settlers came from the USA. For clarification, the USA's presidential system, with its checks and balances, is in contrast to parliamentary systems elsewhere in the world, such as in Germany or Japan. Presidential systems place quite a lot of power with the executive.

Liberia picked up the USA's constitution and its inspiration---the Republic's motto is even "The Love of Liberty Brought Us Here"---but also brought to West Africa the contradictions in the USA system. Namely, slavery and racial hierarchy. Americo-Liberians mistreated native West Africans and subjected them to domination, enslavement, etcetera. This was despite being a democracy. Liberia also developed a partisan 2-party system, much like the USA would, with Republicans and True Whigs. Most all of these factors are inherent to a presidential system of democracy. The Americo-Liberian settlers brought them to Liberia, and can't really be blamed for doing so since it was all they really knew about. Virtually every aspect of the Liberian government was thus modeled after the USA, even though the USA was more or less ambivalent about its only settled colony. Heck, the capital in 1847 was named Monrovia (as in...James Monroe).

So there's an example of the US being an inspiration for another democracy. That democracy did not last very long or prove very stable, by the way. I may elaborate on why or how later, in a second post, but I don't think its necessary since that's not what you're asking about. Instead, I'll probably post the gist of how the presidential democratic system championed by the USA worked after African decolonization in the Cold War.

Sources: Konneh, A. (2011). Multiparty Democracy in Liberia: The Historical, Cultural, and Social Factors That Explain the Nature of Political Competition. Liberian Studies Journal, 36(2), 29-71.

Mbaku, J. M. (2010). Constitutionalism and Governance in Liberia: Where is the Liberty that Brought African-Americans to the West African Coast in 1820?. Liberian Studies Journal, 35(1), 55-97.

VAN SICKLE, E. S. (2011). Reluctant Imperialists The U.S. Navy and Liberia, 1819-1845. Journal Of The Early Republic, 31(1), 107-134.