Great question! I'll focus on French Hip Hop (or "le Hip Hop" if you will) which will hopefully create some more general insights on Hip Hop's travels. It's not meant as an overview over French Hip Hop which would be its own book but rather a focus on its early US connections - Hopefully others can add more on the German and Japenese example which would be super interesting.
[Edit: Intro] I'll add some general influences that I'll trace below with the French example:
Hip Hop culture: Not only Hip Hop music was influential - it was the whole package. In the late 70/early 80s Bronx and Harlem parties would include DJ's, later rappers and b-boys/breakdancers, with graffiti and fashion forming essential aspects of the art form. In the three countries you mention where Hip Hop was already taken up from the mid/late 80s, you can notice how all of these forms were adopted and adapted together.
Check the technique: Sampling formed the basis of the music, creating beats out of breaks and adding sounds. This cannibalistic approach has proven perfect for a globalised world, with Hip Hop on all continents taking up elements of local traditions. Hence you get DJ Krush throwing in Japanese flutes, Mexican group Control Machete adding salsa to the mix, etc. Also rapping, maybe as opposed to different singing styles, has just proven very adaptable to all kinds of languages.
Politics & religion: With Hip Hop originating in African American neighbourhoods in New York, messages of political struggles and ghetto hardships were in the music from the start. These would combine with Afrocentric messages and later Islamic references to reach marginalized people all over. Especially interesting is rap "coming home" to Africa, finding a natural home in African countries with drumming and griot traditions.
Early French Rap & US influence
As in many parts of the world, for many in France the global hit song "Rapper's Delight" was a first taste of Hip Hop in '79. A stronger link was forced from '82 following a tour of Afrika Bambaataa and the Soul Sonic Force, Rock Steady Crew and other rap and breakdance crews. "Bam" is still considered a forefather of French Hip Hop today, with his Zulu Nation's beliefs of "unity, knowledge, justice and peace" providing one main early pillar. Features on Hip Hop culture, fashion etc. started appearing in journald and other media. The "smurfs" b-boys/b-girls group started a movement of DJ's, MC's and dancers. Another mover was the journalist Bernard Zekri, who organised early concerts of US rappers in France and took part in recording some classic tracks - including the oft-sampled "Change the beat" with Fab 5 Freddy.
The French goverment quickly picked up on Hip Hop's growing popularity and courted it as an outlet for French youth to address class issues. The Minister of Culture Jacques Lang at the time invited Afrika Bambaataa to deliver seminar on Hip Hop's potential in France. In 1984 a chapter of the Zulu Nation formed as a result. Lang even declared optimistically:
Even if in the beginning it [hip-hop] drew inspiration from America, I believe it has found its originality here in France. [Aidi, 208]
This French gov't facilitation
"culminated with the creation of the 'Movement Authentique' in the Paris suburb of Vitry-sur-Seine", that included not only French rap and breakdance but graffiti "tagging" on the Paris Métro. As a result of the Ministry of Culture's orchestrated hip hop seminars, a network of community centers were created througout the country as a new youth culture phenomenon in France. [Osumare, 85-86]
Many of the new rappers coming up were from African, Arab, or lower class backgrounds, often with backgrounds in France's former colonies. One major element that appealed was the common reality of oppression connected to a powerful sound. Many rap artists lived in the banlieus (or suburbs) that had often been built outside of larger cities like Paris and Marseille and which helped sustain a racially hierarchical soceity. Messages of hardship from the American Ghettos translated easily to the banlieus. Or better put
Resonances between different kinds of marginalities prompted vigorous adaptation of hip-hop's aesthetics and resistice social agendas in France. ... Grassroots hip-hop culture based in the suburbs of Paris, Marseill, and Lyon, rather than the inner cities of the United States, evolved as a way of creating expressive rejoinders to the system that they perceived as oppressive. [Osumare, 86]
Edit: Forgot to add Le cuir usé d'une valise by La rumeur here, which is a bit more recent (2002) but still I think shows well a group dealing critically with France's colonial past and banlieu present. This shows in lyrics like "the old colonial heritage dominated for centuries, connecting le Havre and its surroundings from the sinister holdings of a ship", and "at least the rats and cockroaches live together in peace, with the vulgar symbols of post-war France" (my transl.). The song also showcases the trauma of slavery shared by Afro Americans and African French in different ways.
Another major element was the Afrocentric thought put forth by American rappers in the 80s (Bambaata looms large again), and taken up again by later crews like A Tribe Called Quest (tied to the Native Tongues crew) in the early 90s. Many of the main French rappers were from or descended from the Maghreb, with the bloody Algerian war of decolonisation one especially traumatic background. Probably the most famous rapper (also internationally) at the time was MC Solaar, of Senegalaise origin. The dandy du rap helped mainstream the genre, bringing it to the charts. Maybe more interestingly, he also connected to one of rap's origins, the "griot tradition of French West Africa. In addition to his rap style he would also incorporate African instruments into his songs. Then again, Solaar would also sample more "traditionally French" music - just have to put in "Nouveau Western" here, both for it's amazing sample of Serge Gainsbourg and Brigitte Bardot's "Bonnie and Clyde", and for it's lyrics that are interestingly critical of the American way of life (and the Western).
Many early (and later) rappers aimed anti-establishment raps at the French nation and its institutions -- lyrically recalling the work of more politically minded US rappers like KRS One or Public Enemy. Interestingly these same rap groups had often taking to referencing civil rights leaders like Malcolm X, and through this would increasingly reference Islam or Islamic countries (I go into some more detail on this in this earlier answer edit: and also highly recommend the answer by u/hillsonghoods in the same thread). Hisham D. Aidi sees influence of Islam spreading through Hip Hop to France and other regions - but also strengthening how the music resonated with French youths of African origin. A shift is usually made out somewhere from the late 80s from the more "conscious" raps following Zulu Nation ideals, to more hardcore artists like Assassin and Suprême NTM . Once more a borrowing took place, this time taking more from US gansta/hardcore rappers becoming hugely popular at the time. Fun fact: Around this time France also came to be the 2nd largest Hip Hop market after the US.
Another major development came in the mid-90s with a stronger fear of Islamic influence through rap, stoked by right wing politicians. Politicians like the Front National leader Jean-Marie Le Pen warned of a dangerous art form coming in from Algeria - keep in mind the traumatic memory of the Algerian war for French and Algeriean people; and that Le Pen had served in that war. In his recent and fascinating study of music and Islamic youth culture Hisham D. Aidi notes that
Hip Hop came to encapsulate France's anxieties about both American domination and Islam. ... The 1994 Carignon media law required that a minimum of 40 percent of musical programming on radio stations be by French musicians and 20 percent by contemporary French artists. This quota acutally encouraged the expansion of French Hip Hop, as FM mega stations shifted their playlists from Anglophone dominated pop and rock to rap, where they thought the French-language quotas could be met more easily. French record labels began expanding their hip-hop portfolios and sending scouts into the banlieus, looking for talent. And, ironically, a law designed to protect "French identity" from Americanization helped promote the French form of an American genre, one that now amplifies the voices of religious and ethnic minorities seen as a domestic threat to national identity. [Aidi, 208]
So we can note the overall influence of US rap through messages of Afrocentrism, a shared struggle (of marginalized Afro-Americans and French Africans and Arabs), and Islam. But also in the 90s a stronger move away from the American roots towards French language and different sounds in French rap, expanded through its increasing rise via radio and other media. The #1 track I can think of summing up these various tensions would have to be Assassin de la police's mesh of KRS One and Edith Piaf.