I will take a stab at answering this, although attempting to explain almost 20 years of international geo-political warfare in a cohesive answer will require a degree of summarisation.

This is a tricky question as the breakdown of the wars into the periods known as coalitions is not completely accurate. Initially, the alliances of countries arrayed against Revolutionary France were not referred to as coalitions, this labels were appended on those conflicts later. Another problem is that in some cases the coalitions did not necessarily end in defeat before changing to another - for instance the Fourth Coalition was more a case of new allies (Prussia) joining existing ongoing warfare (the Third Coalition) even though some allies in the earlier coalition were defeated.

Also, there was only one constant "allied" nation within all of the coalitions and that was Great Britain who, apart from a brief period of peace from the spring of 1802 to the spring of 1803, was constantly at war with France for the whole period. The coalitions are best seen against this backdrop with alliances forming to oppose France, not for plans of grand coalition security, but in reality for the perceived interests of individual states. For instance, the Third Coalition was an alliance of Russia and Austria opposed to Napoleon's actions in areas they believed to be near their own interest - in this case the Germanies and Italy respectively. Whereas the Fifth Coalition was formed as Austria attacked France to seek the recovery of territories lost in the 1803–1806 War of the Third Coalition, hoping to succeed while France was distracted in the Iberian Peninsula, where France was already at war with Britain, Spain, Portugal, et al (France had shifted about 100,000 troops out of the Germanic territories which probably helped Austria feel they had an opportunity). The Sixth Coalition which drove France to defeat in 1814 was a result of Napoleons disastrous invasion of Russia in 1812, and this weakening of the French army allowing Prussia and Austria to break their imposed treaties with France, and join with Russia, the United Kingdom, Sweden, Portugal, Spain et al who, again, were already at war with France.

So it might answer your question to think of them not in terms of coalitions, but as shifting alliances brought about by opportunistic changes in the geopolitical picture of a war-torn continent. British subsidies for allies willing to form armies to fight Napoleon was also a feature, allowing previously defeated nations the ability to partially fund their rearmament which, combined with the costs of Britain's own armed forces and other needs, contributed to driving the British National debt to almost £680 million - twice its yearly GDP - a staggering figure for the period.

Forrest, A. "The French Revolutionary and Napoleonic Wars" in Early Modern Military History, 1450–1815 (Palgrave Macmillan, 2004), pp. 196–211.
Grab, A., Napoleon and the Transformation of Europe (Palgrave Macmillan, 2003)
Watson, J.S. The Reign of George III, 1760-1815 (Clarendon Press, 1960)