The biggest part obviously was the French Army, recruited from all the directly annexed French lands, so besides proper Frenchmen we also find Belgians, Dutch, Italians and Swiss. The rest of the army was filled by vassals and allies. "Three quarters of the army were composed of nations whose own true interests opposed the beginning war.", according to Count von Wedel of the 9^th Chevaulegers. But even so, with Napoleon's presence and under the conditions of battle they all fought bravely for the cause.

After the French, the biggest contingent was Polish, partly of the Duchy of Warsaw, which Napoleon created in 1807 out of Prussia's share in the Third Partition of Poland, and partly directly in Polish Regiments in the French Army; all in all about 95.000 men under Marshal Poniatowski. The Polish were keen allies for Napoleon's cause because they hoped he would free the Polish regions under Russian control and reunite Poland. Many were experienced fighters, having fought with or alongside the French Army since the Revolution, but for the 1812 campaign the Duchy of Warsaw couldn't provide the required numbers properly, which lead to cutbacks in training and equipment and the recruitment of unfit men.

Then we have the Italians from the Kingdom of Italy under its viceroy Eugène de Beauharnais (the Italian throne also belonging to Napoleon) and the Kingdom of Naples under their king, Marshal Murat. Alltogether about 45.000 men, the Italians organised and trained according to the French military standards, the Neapolitans more of a badly trained bunch who started deserting as soon as they left home.

Following, we have the Confederation of the Rhine. The Bavarians, who had already proven trusty allies and joined France in many military endeavours, provided 24.000 under General Deroy, Westphalia 17.000 under their king Jérôme Bonaparte and the Saxons also had 20.000 to contribute, including some of their best cavalry. Those big three had their own command, while the rest of the German states were directly under French command. All in all about 130.000 German troops.
In detail, they were 12.000 from Württemberg, 8.000 from Baden, 5.000 from Berg, 4.000 from Hesse-Darmstadt, 2.800 each from Frankfurt and the Ernestine duchies, 2.000 from Würzburg and lastly 3.800 from Nassau and all the tiny states (like Hohenzollern-Sigmaringen or Liechtenstein) it had to co-organise.

Not to forget the two Spanish volunteer batallions from Joseph Napoleon's Regiment under Colonel Doreille, the 5.000 Portugese (and also Spanish) under the marquis d'Alorna and the 3.500 men from the Illyrian Provinces.

The Prussians had to provide 20.000 men, about half of their total army, as auxillary under command of the French Marshal MacDonald, while the Austrian auxillary of 35.000, about a fifth of their total army, could remain under Count Schwarzenberg's direct command.

Most of those troops were part of the Grande Armée that went all the way to Moscow, through the bloody battles at Smolensk and Borodino, and back in a disastrous retreat. Notably also that Napoleon recruited cavalry from pretty much all of them for his I Cavalry Corps under Murat, a gigantic cavalry-only part of the army meant to intimidate the enemy by its sheer might, but mostly ineffective and a decisive reason for the almost total loss of Napoleon's experienced and properly trained cavalry he could have needed at Leipzig.

Unfortuantely for Napoleon, his biggest enemies in his army had a rather relaxed time. The Prussians under Marshal MacDonald were securing the northern flank and were so unbothered by the Russians that during the summer officers would regularly go swimming in the Baltic sea.

Schwarzenberg meanwhile was tasked with securing the southern communication routes in today's Poland and Belarus areas, for which he had the command not just over his Austrians but also the Saxons under Reynier, who was not just there as general but also French advisor and spy. Supposedly there was an agreement between Russia and Austria that both Schwarzenberg and Tomasov's Third Army would try to avoid each other as much as possible while still keeping up the appearance for Napoleon. Certainly the South only saw one major battle at Gorodetchna in late July, which ended with an Austrian victory that cost 4.000 Russians and 5.000 Austrians and Saxons their lives, which represents about 60% of Schwarzenberg's total losses and impressed Napoleon enough that he requested Schwarzenberg be promoted to Field Marshal. At the end of November they retired to their winter quarters at Byalistok under a verbal ceasefire agreement with the Russians, effectively ending their part in the campaign.

That is the organisational integration at least. If I misunderstood and you wanted more about social interactions between the different nationalities I'm sorry and I can try to provide that as well.

Main source for the numbers:
Adam Zamoyski's Moscow 1812: Napoleon's Fatal March (2004)