score:6
Using information about the Orient Express en de:
What is the average travel speed of a train going from one European capital to another in the late 19th century?
German wiki says: 48 km/h (over the whole length of the Orient Express)
Β Vienna/Paris
was served by the Orient Express, duration: 2 nights + 1 day.
Vienna/Berlin
Accoring to this newspaper article about the (much later) Berlin - Vienna connection "Vindobona" there was a Berlin - Budapest variety of the Orient express. FWIW, the Vindobona Berlin - Villach had an average of ca. 90 km/h in 2014.
During WWI, the Balkan Express is quoted with an avg. of 32 km/h for Berlin - Istanbul. It also had a section going Berlin - Vienna.
Wiki on Orient Express mentions that there were Berlin - Vienna connections.
Would such trains also travel during the night?
The Orient Express was operated by the "Compagnie International des Wagons-Lits", so, yes, night trains.
Paris - Vienna is 1360 km and took "2 nights and 1 day" (Wiki), avg. of 48 km/h would mean 28.5-ish hours.
Berlin - Dresden - Prague - Vienna is about 800 km, so it took probably something between 16 and 24 h (for 30 - 50 km/h avg), thus maybe doable without night train, maybe not.
Upvote:4
In Jules Verne's novel Around the World in 80 days (1873) the protagonists travel around the world in 80 days in 1872, going by steamships for most of the journey, but various railroads for most of their travel on land.
Phileas Fogg planned his journey using the timetables & schedules published by various steamship and railroad companies, and it seems quite possible that Verne actually consulted the schedules of the railroads in his novel. Verne was inspired by non fiction articles explaining that it was now possible to travel around the world in just a few months.
So the times, distances, and speeds of various train trips in Around the World in 80 days (1873) should give a good general idea of average train speeds in 1872. Other works of fiction by careful 19th century writers should give fairly accurate ideas of the speeds on various European train lines. (Some writers are very careless about such details).
No doubt there are online railroad history sites where you can ask questions about specific lines at specific times.
Upvote:4
Since the other answers don't answer the question about border crossings, I will answer it: They wouldn't have introduced much of a delay at all. Due to the increased international travel brought about by railroads and the lack of any long wars (or really any major European wars between 1871 and 1912), passports were not required for most border crossings from the 1880s until the outbreak of World War One. The EU's institution of passport-free borders is just recreating the status quo ante 1914 for the movement of people. There would of course have been some restrictions on goods, but cargoes tend to travel on different trains from passengers, so there would not have been much inconvenience to passengers from that.