Okay, the actual historical answer will probably get buried at this point, but why not. It's awesome to have a black history question about something besides slavery!

Black Americans were actually very prominent in 19th century efforts to evangelize Africa. Before the Civil War, when true missionary efforts to Africans were less enthusiastic, the vast majority of black missionaries were sponsored (financially and institutionally) by white organizations. Famous early evagelists like Lott Carey sailed the Atlantic on ships of the (white-founded, white-run) American Colonization Society, which mobilized to establish Liberia as a colony for emancipated American slaves. Most of these missionaries ended up serving the Liberian community, although a few, like Carey, seemed to view it as a place from which to reach out to local Africans.

The high point of black American missionary efforts to Africa was after the U.S. Civil War. Initially, some white mission boards continued to be huge proponents of supporting black missionaries. White America's "gift" was the way it had divorced newly-arrived slaves from separate identities and lumped them all as black Africans. Black missionaries were seen as uniquely able to minister to Africans and physically equipped to handle tropical diseases better (which was not overall true, since they'd been raised in America).

But in this period, especially after 1870, black churches also mobilized to support their own missionaries. Jacobs and Martin list a few reasons for this date. First, black churches had to absorb the impact of emancipation. Second, the end of Reconstruction and ongoing racism in the U.S. convinced some black Americans that repatriation to Africa was still the solution. Third, the racist attitudes of many white missionaries were becoming ever more apparent to black missionaries sponsored by those mission boards and societies.

In the decades around the turn of the 20th century, however, black American missionary presence in Africa basically fell off a cliff. This was certainly not due to lack of interest among black American Christians.

The Civil War and its aftermath had had the result of creating closer trans-geographic ties within black church traditions than white ones (which had frequently been divided by viewpoints on slavery). Initially, this was to their great financial benefit, having a wider basis of support. But the end of Reconstruction and the rise of Jim Crow laws were disastrous for southern blacks financially as well as in all the other ways, a catastrophe which reverberated in black church denominations. Very little money could be spared on missionaries.

Sandy Martin suggests that black missionaries were also confronted with a unique problem: the psychological costs of understanding black Africans as spiritual and physical kin, yet perceiving themselves as superior and civilized (black American missionaries' descriptions of black Africans are as colonial as whites': backwards, heathen, dark continent, etc). There isn't really evidence to suggest this became a particular problem that reduced black American missionary presence after 1890, though.

The larger problem was white mission boards' racism. They refused to send black missionaries of their own--and they even outright impeded the efforts of black mission boards to sponsor evangelists to established missions, for several reasons.

Evangelization of Africa became closely tied in with an mindset of imperialism and colonization. White Europeans very nervously watched African rebellions, especially in eastern Africa, and worried that black Americans would "naturally" side with black Africans. They pressured the mission boards to quit sending black missionaries. The rise of pan-Africanism and Black Nationalism in the U.S. (think Marcus Garvey) only added to white Americans' desire to add to (or give in to) the pressure. There was also a good dose of straight-up white racial superiority. Black missionaries, they argued, simply weren't good at their jobs. Why take the risk that they would incite or contribute to rebellions, if they weren't even any good?

By the 1920s, black American missionary presence in Africa was tiny. With later waves of American evangelization efforts (including ongoing Pentecostal efforts), colonial rebellions couldn't be a real issue as the trajectory towards widespread independence was well underway. However, there were a couple of big roadblocks for black American involvement. First, financial. Black churches were simply not as well-funded as white ones, thanks to ongoing American institutional racist policy (in the north as well as south). Second, recruitment. Mission societies trawling for potential missionaries targeted young people--at colleges. In midcentury America, this was predominantly a white thing. (And when it wasn't--the HBCUs--one imagines it was easy for white groups to overlook them).

As to the short-term "mission trips" (dig a ditch, rock an orphan) so popular today among certain Christian groups, that is a question more for /r/AskSocialScience or /r/AskAnthropology than a history sub.

The only two major religious groups that require evangelism are the Mormons and the Jehovahs Witnesses. For everyone else, it's choice and therefore requires a relatively comfortable socioeconomic status to engage in.

Mormons are overwhelmingly white(85%) and while JW's are more evenly composed, their evangelism has a different character and relies more on local groups rather than foreign missionary work.

As a follow-up question, how have people tended to pay for mission trips in the timeframes your looking at? Were there economic factors that allowed (or encouraged) certain groups to become missionaries more than others?

Recommented to emphasize timeframe being the same as the OPs question.

This may or may not be directly relevant to your question, but I think some of this racial element being equated to an American sense of Christian mission traces back to the colonial era. Jill Lepore's book "The Name of War" examines King Phillip's War and concludes that New England colonists tied Christianity strongly to race, by creating a category of dangerous "others" (Native Americans) that were defined by race and religion. The non-whites, non-Christian "other" was seen as an existential threat, and this sense of cultural identification with "White" and "Christian" as things that needed to spread to stamp out the "other" became part of what it meant to be an American colonist.

Obviously this is very interpretive, but she backs up her conclusions with a thorough analysis of the language that colonists used when talking about the war. It might be an interesting read as a starting point to answering your question. There might be better sources on this particular issue, in all fairness, I'm not a religious historian.

What period are you looking at? Because there has been big changes in American religion particularly with how it relates to culture. There's also been changes in Christianity in Africa.