If you already know at what dates you'll be traveling, then yes, there is an advantage. All trains between London and the continent, all decent trains between Paris and Brussels, and all convenient trains between Brussels and Amsterdam are high-speed trains with compulsory bookings. The more in advance you book, the less risk there is that the train will be full, and the less expensive the fare will be.
You can get a print-at-home ticket-and-reservation for all of these journeys over the Internet. Shop around: sometimes international fares can be different if you buy them from one country or the other or from the train operator. Print-at-home tickets are nominative, you can't resell them to someone else and you'll need to show your passport or other identity document to the ticket collector.
For shorter journeys on regional trains, there is no reduced fare and no reservation, so there is no advantage to buying a ticket in advance except not having to queue at the station. Beware that in France, train ticket selling machine require a credit card with a chip or coins; there is no surcharge for buying a ticket at a window but there can be a queue. In the Netherlands, ticket machines require a Dutch banking card or coins and there is a surcharge for using a ticket window.