When reading about the Napoleonic Wars, one thing that stands out is the fact that Britain was Napoleon's main antagonist throughout the successive wars. However, direct engagements between French and British troops were much rarer than, for example, confrontations with Austria.
Since Britain is much closer to France, one would expect frequent clashes either in the English Channel or via expeditionary forces. Napoleon dreamed of invading Britain and dealing with "the nation of shopkeepers" once and for all. The obvious reason this didn't happen is that miles of coast are much easier to defend than conquer (Omaha Beach comes to mind). After the Royal Navy sunk the main parts of the French fleet at Trafalgar, an invasion of the British Isles was next to impossible. Any invasion would have had to rely on favorable weather and hope to not be detected by superior Navy ships.
After the French fleet was dealt with, Britain blockaded French ports in and around the Mediterranean and seized French vessels. Britannia ruled the seas and had a world-spanning empire. The first Industrial Revolution meant that British productivity1 and industry outpaced the rest of the continent (apart from the Netherlands)2.
Why did Britain not invade France (earlier) and put an end to French wars on the continent?
Britain was a naval power akin to the United States in the 20th century. Her wealth and dominance came from a world-spanning trade system that allowed for the global flow of goods to and from Britain (denoted in GBP) and an industrial base that was technologically superior.
This meant Britain had a lot of money, to put it crudely.
If Britain wanted to fight France, it had multiple options:
It could raise a large continental army and fight on the continent itself, but at great risk. It would have been a huge logistical undertaking to supply an army across the water in enemy territory. The clashes between continental forces in the Napoleonic Wars resulted in the biggest loss of human life at least since the Thirty Years' War in Europe3, making deploying a full continental army unattractive. Britain deployed troops in Portugal during the Peninsular War, landed an expeditionary force to retake Holland (which failed), and aided the final coalition as Napoleon was already severely weakened after losing in Germany4.
Britain had some experience with continental armies in the form of the troops used by the East India Company during their conquest of India. However, the key difference was that in India, the EIC was using Indian troops (sepoys) led by British leaders5, rather than troops comprised of a majority of British soldiers.
Britain had a much more attractive option on the table:
Use its dominance of commerce and the sea to prop up France's enemies on the continent and have them fight the French armies or threaten the French homeland. In various instances, it outright paid Russia, Prussia, and Austria to fight France. It used diplomatic and economic ties to bolster the enemies of France. It supplied weapons and supplies to various allies all over the continent and supported insurrections against French rule6. Britain's strategy resembled modern Lend-Lease strategies. All of this meant that Britain had no desire to raise a large army herself but rather have her allies assume the human cost of the fight against Napoleon and the destruction to their territories. The fact that the British Isles were never under direct attack meant that they could keep expanding their trade networks, grow their economy, and keep supporting its allies.
Napoleon saw some of this and tried to weaken the British economy through a continental tariff regime but never fully succeeded. One could argue that a massive economic downturn might have ended Britain's policy and led to a collapse of its ability to support the various coalitions, but the higher productivity and her colonies meant she had ample resources.
Britain spent a huge amount of money on supporting her allies, but it was a cheap bargain compared to the cost of fighting herself.
https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=8c94b8f3de290cb3190f5704df516081e197facc ↩
https://econ.columbia.edu/wp-content/uploads/sites/18/2021/11/bns_malthus.pdf ↩
https://www.napoleon.org/en/history-of-the-two-empires/articles/napoleon-the-dark-side-the-human-cost-of-the-napoleonic-wars-3-min-read/ ↩
https://www.rusi.org/explore-our-research/publications/commentary/coalition-diplomacy-french-revolutionary-and-napoleonic-wars-great-leap-forward ↩
https://www.worldhistory.org/article/2080/the-armies-of-the-east-india-company/ ↩
https://en.wikipedia.org/wiki/United_Kingdom_in_the_Napoleonic_Wars#:~:text=Through%20its%20command%20of%20the,Peninsular%20War%2C%20Britain%20played%20a ↩