Some Medieval sources, say that Slavs, or otherwise known as Sclavonians, were slaves(servants) from different origins of the Byzantine Masters.
The origin of the word is just "Sloboda", a village or place to live in ("s-" as a prefix, "life" as a root of word, this gives a village).
The theory, which makes them a big branch of IE people, migrating in different direction all over Europe, is not correct.
Although you're right about the whole Europe, there are many facts of archeology about Eastern Europe, Baltic Region and Northern India, which would help to trace our origins and roots. I tried to clarify it in the neighbour thread,
https://www.eupedia.com/forum/threads/28906-New-migration-map-of-haplogroup-R1a1a
There are many artifacts which proves that ancestors of Northern Indians, Balts and Slavs started from the agricultural areas near the northern bank of the Black Sea, when it was a freshwater lake, then came to the North to establish a Dnieper-Donetsk agriculture, which includes cities Kiev and Zhitomir ("Key" was a name of the knight who was the governor, "zhito" means wheat), and then, — as bricks and big white "Russian" stoves has been invented, — continued to move to the North, i.e. to the sourthern bank of the Baltic Sea, when others came to the East, i.e. to the country now known as Russia.
There was a huge role of Scandinavian people to establish a state. "Varyagi" or just warriors, came from the far far North, probably from Sweden. They were part of Kiev city inhabitants and known as "Varangian Guards". Living in Russia you could read folk tales about them. Some names, including Igor (Ingvar) and Olga (Helga), are Scandinavian ones.
As years passed by, most of Russians and Balts in those areas became a mixture of Slavonic and Finno-Ugric people, except for Estonians who are 100% Finno-Ugric people. There are many folks inside the country who still speaks in Finno-Ugric languages, especially in villages. Mari music is a gem for those Russians who can play musical instruments.
Northern India was conquered by people who started from the same region, the fruitful, rich, fertile banks of the Black Sea. Both Krishna and Arjuna spoke in "Old Church Slavonic", also known as... Sanskrit.