I am not able to discuss in details the spread of Fascism through the various European Nations following its establishment as a form of government in Italy. What I can say though is the fact that the "national" character of a political doctrine - of any kind - didn't really prevent its spread to other Countries.

Nationalism itself was common to any Country I can think of in the late 19^th early 20^th Century: this evolution was perhaps stronger in Nations such as Italy and Germany were the recent process of unification allowed for the ideals of Risorgimento to mix with the new modern take on the Nation. In Italy the myth of the Nation could take its strength from the ancient glory of Rome, the cultural prominence of Italy during the Reinassence, the republican and individualistic values of the Comuni - city states - but also the universal values of religion, language; the national call of the glorious enterprises of Garibaldi and the Independence Wars, the almost mystic "nation religion" of Mazzini; and more along the road ideas of racial superiority and the myth of the Mediterranean as an "Italian sea" - for example the nationalist demands for WW1 spoils went as far as to include regions of Turkey and Egypt to be put under Italian control or supervision.

But other Countries had their Nationalisms too: the French even had their own word for it! - jokes aside I do not know much of French Nationalism, but I know that it existed and certainly did not see itself as an imitation of any other Country, but taking inspiration from a set of "national values", proper of that specific Country.

As for Nationalism - which never aspired to be an international movement, as Marxism did - it really grew on the concept that any Country could have its own National values, i.e. its own Nationalism.

Under this aspect, it is relevant to observe that a common trope of the Nationalist literature was the "character of the nation, or the character of the people": in Italian political polemics it is fairly common to see the rejection of parliamentarism framed as rejection of some foreign import, either from France or Germany, according to the focus of the writer. Each Country had its own national values and therefore had to have its own form of government, politics, etc. - in this vein you see the rather widespread idea that while the British could be content with a Parliament, the Italians needed a strong man, looking back to the Dictators of old Rome, or the Cesars.

This should help you understand that there was nothing to stop the spread of Nationalism through Europe and every Nationalistic Movement claimed or aspired to identfy with its own idea of "a Nation" while every one of them identified with the idea of "Nation" - and thus it is reasonable to group all of them under the same denomination while acknowledging that every one of them was individual.

A similar case can be made for Fascism - with the observation that the spread of Fascism may have been less succesful than that of Nationalism. In a certain sense, Nationalism never aspired to rule the country exclusively through a Nationalist Party; in Italy, most of the nationalist movement flowed into the newly established Fascist Party as early as 1920-21, bringing with them a good chunk of their political ideals and a far smaller contribution to the actual ruling establishment (the Nationalist Federzoni held a relevant ministry in the first Mussolini Cabinet). In fact the most relevant contribution of the Italian Nationalists was of intellectual nature: Prezzolini - one of the founders of Italian Nationalism - saw his action as a plight for action and reforms aimed at the true forces of the nation: the industrial owners, the active honest workers, etc. - against the old passive, ineffective political establishment; thus Prezzolini did not want a Nationalist Party, rather to spread the ideals of the nation among the good forces of the Country so that they could band together and bring forward a process of renovation; in alternative or even opposition to the Party System.

Still the ruling classes, while clearly influenced by those ideas, needed instruments - both legal/parliamentary and illegal/violent - to control the percieved socialist threat in the years 1919-21; and here you can one of the reasons why Fascism proved more effective, since it supplied both those things, and why it was able to rise and gain power in Italy.

But this process was often not completely developed in other Nations - or did not happen at all - as you can see for example in R. Paxton Anatomy of Fascism.

So that it is not really true that Fascism spread internationally that easily: other Nations went with the safer route of a Conservative/Nationalist/Military Party establishing an authoritarian government - think for example of Portugal, Hungary and Romania. Others like Germany developed a "form of Fascism" that grew from their - obviously - peculiar set of values to a maximum of radicalization that was under many regards an unicum in History.

An overview of this process and the connection of Nationalist movements with the Fascist ones throughout Europe would be too much for one answer; but I hope to have give a decent enough summary of why Nationalism and Fascism could develop in many different Countries without having any aspiration to intra-national unity of intents.

Ideas such as anti-parliamentarism; rejection of old politics; a claim to represent the true values of a nation, a people, a "blood"; the myth of effectiveness against bureaucracy, action against empty words; leaderism... were general enough to find a good soil in almost every Nation. The strenght of the traditional institutions, the presence of safer alternatives for the ruling classes, the ability of the leaders of the Fascist movements, were the decisive factors in the establishing of Fascism and the form it took.

None of these aspects was international: every Fascism could be its own Fascism, and it had to, since it was, as Nationalism, strongly rooted in the Idea of Nation. Yet it is convenient and useful to see and understand the connections - that contemporaries too were able to understand to some degree - between these various movements and to group them together under the name of Fascism; as it is done for the manifold Nationalist Movements.