Germany did in fact ask their allies to deport "their" Jews - but the manners of such requests changed with time and depending on the specific relations with those nations. Since you chose Italy as an example, I'll address that (not that I could address the others in any detail).

First, the idea that Italy introduced a racial legislation due to German pressure is not really something everybody agrees upon - even if it plays well with the common narrative of a Mussolini reduced to Hitler's "lackey", and somehow takes away part of the responsibility of the Fascist Regime. It is a popular view – and you'll find it mentioned in various works on the subject, but there are very few direct or indirect confirmations that the Nazis applied direct or indirect pressure on the Fascist authorities (G. Ciano angrily remarks in his diary soon after the introduction of the racial laws how “The Jews were accusing them of doing as the Germans told. False! The Germans never spoke of that.”).

What they did – and gets probably conflated with the racial laws of 1938 – is demand, after 1943 and more strongly with 1944, or to an extent order the Italian authorities to surrender the Jews for the purpose of their extermination. As a result, of the 7,495 Italian Jews collected by the Nazi occupation forces and the collaborating Salò Regime for deportation in the early months of 1944, 6,885 died there.

I am talking more general of "racial laws" since the context for the development of those laws within Italy is that of the renewed interest for colonial adventures during the 1930s, with the “pacification” of Libya and the conquest of Ethiopia. That is, the antisemitic laws of 1938 – consistent as they where with the German Nuremberg Laws, and conveniently falling within the time frame of the Italo-German alliance – appear to be more a continuation of an ongoing “segregation-like” policy, originally targeted to the African natives within the Italian colonies.

According to R. De Felice, the choice of switching the Regime's attention towards racial themes came in the context of the general dissatisfaction of the Regime's hierarchy, and of Mussolini especially, with the Italian people, their inadequacy and resistance to a true and complete fascistization. Mussolini, paraphrasing his own complaints, had given them an Empire – and instead of proving worth of it, the Italians were revealing all of their traditional, natural flaws. The formation of a new fascist generation was going to require a strenuous effort (and in this sense, it's very likely that the National Socialist example may have influenced the many smaller and larger leaders who chose “to re-establish their fascist purity” - quoting G. Bottai by memory – by going on a mission in Germany around 1938) since the current Italians were certainly not up to the task (and by the mid 1930s Mussolini was already certain of the inevitability of a future European conflict, by 1942-43 at most).

Soon after the conquest of Abissinia, a prominent Fascist Gerarca and leader of the “intransigent” current, Roberto Farinacci, had traveled to the new colony and wrote back to Mussolini (April 24^th 1938):

Our [con]nationals are not well guarded and give the natives a far less than imperial show. Many ][unfortunate] episodes are caused by their behavior, either gross, indecent or offensive [...] absolute lack of understanding and respect for the traditions of the natives, dreadful attitude towards the native women, loan demands towards well off natives, indecent ways of living and dressing in comparison to the natives.

How could the Italians claim mastery of other peoples if they couldn't even stand tall above... well... As Mussolini himself explained in a private speech of October 25^th 1938:

[…] when a people takes conscience of its race, it does so in contrast to any other race – not one alone. But we did so only in opposition to the “Camites”, which is to say the Africans. The lack of racial conscience has had severe consequences in the Amara [a region of Ethiopia – likely a synecdoche]. […] when they saw that Italians went more raggedly than themselves, that they lived in “tuculs”, that they took away their women; [the natives] said to themselves: this is not a race that brings us civilization! […] To preserve the Empire it is necessary that the natives have absolute, undisputed understanding of our superiority.

This core concept had found its way in Mussolini's mind somewhere during the previous years – and we know that with some certainty since, according to his habits, Mussolini had favored the publication of racist and antisemitic works since around 1936 (that is before the Italian relations with Germany grew close enough to warrant a direct ingerence – as I mentioned, there were intellectual contacts and growing ideological contiguity, for instance with “experts” exchanging their views on the matter on specialized press, but there is no evidence of political or diplomatic pressure during 1938-39), for example Paolo Orano's “The Jews in Italy” published in April 1937. There is little doubt that the antisemitic turn around 1938 had a large portion of instrumentality in Mussolini's mind – but that did not mean it was unplanned, nor entirely unwelcome. Rather, the view that ideological positions had to serve as instruments was one of the traits that more strongly separated Mussolini from Hitler. As Mussolini had attempted to explain in a letter to the newly appointed German Chancellor in March 1933:

Every regime has not only the right but the duty to remove from places of authority any element non entirely trustworthy, but to this end it is not necessary, rather it could be damaging, to bring on the racial ground – Semitism and Arianism – what is instead simply a measure to protect and develop the revolution.

What then if the racial angle appeared to be more effective than Mussolini had believed at first? The decision to take a different approach to the problem, one that strengthened the relation with the new ally, established a further mean of pressure over the Italian society, tried to secure new basis for an identification of the “true Fascist Italian”, was certainly not out of character for Mussolini. At the same time, it did remain always an instrumental one – Mussolini's own, as I believe along the lines of De Felice – but dictated by Mussolini's approach to political problems in a “non-ideological” manner. For this reason, the Fascist authorities were content of applying to the Italian Jews a legislation similar to that devised for the African natives – albeit with a few extra provisions.

In the Racial Manifesto of July 14^th 1938 it was stated:

The concept of race is a purely biological one. It is based therefore on different considerations than the concepts of people or nation […] but at the basis of the differences between peoples or nations there are differences of races.

It is time for Italians to declare themselves racist. All the work done by the Regime so far is indeed racism. […] The question of racism in Italy has to be addressed from a purely biological point of view, without religious or philosophical connotations.

Jews do not belong to the Italian race. [They] represent the only race which was never assimilated because it is composed of non European racial elements.

The actual text of the Laws (October 6^th 1938) went into further details:

The Grand Council of Fascism, after the conquest of the Empire, proclaims the urgent actuality of the racial question and the need for a racial conscience. [...] The Jewish question is only the metropolitan form of a general issue.

The Grand Council of Fascism reminds everybody that international Jewry […] has been the soul of anti-fascism on every ground […] All anti-fascist forces are led by Jewish elements.