The Yorkists and Lancastrians were two political movements during the War of the Roses. They formed informally after 1399 in response to the overthrow of Richard II by his cousin, Henry of Bolingbroke, 2nd Duke of Lancaster. The whole thing was a constitutional issue: three men had good claims to be Richard's successor, but Henry wasn't the recognised Heir Presumptive and had been banished by Richard before returning to take over.
The Lancastrians argued that Henry was the rightful heir, and that even if he wasn't it doesn't matter since he took over via Right of Conquest (rendering the legitimate government so useless it is no longer fit for purpose). The Yorkists meanwhile argued the conquest was illegal and he still wasn't the rightful heir.
The day before Richard II's overthrow, he had a number of male heirs to the throne. Personally I can find ten cousins and an uncle who were alive at the time, with some in the Portuguese Royal Family. However, there are three men who have the best legal claims.
To add to this mess, you have the typical infrafamilial marriages of the time, so several generations later you have the future Richard, 3rd Duke of York being descended from three of Richard's uncles.
Where does Henry VII fit into this? Of course he's not born yet, but his claim derives from John Beaufort, 1st Earl of Somerset, who was Bolingbroke's oldest half-brother. Problem is, the Beaufort children were all illegitimate due to a constitutional problem - their father John of Gaunt had several children to his mistress, then married her after the fact.
Henry grew up in a household that supported the Lancastrian cause, and was still vaguely related to the Lancastrian monarchs. When the legitimate Lancastrian lineage was extinguished, the movement turned to the Beauforts and declared Henry the rightful heir (over James III of Scotland, his cousin). Yes, he wasn't legitimate, but since he took over via Right of Conquest he had that legal loophole sorted out, with the marriage to Elizabeth of York to shut up the naysayers.
Summary: The Lancastrians were a political movement - not a family - and Henry was a distant relation of the Lancastrian royal house so became their heir.
Henry Tudor, who began the dynasty as Henry VII, was Lancastrian more in spirit than in name. His mother Margaret was a granddaughter of John of Gaunt, Duke of Lancaster, a son of Edward III. The line of Lancastrian Kings descended from John’s son Henry IV. John was married 3 times, the first two being political marriages, the third being to his long term mistress, Katherine Swynford. The couple had several children out of wedlock who were given the surname Beaufort after one of Johns castles, and were later declared legitimate by Richard II. Beauforts were prominent magnates under the Lancastrian Kings. Henry VII’s mother was a Beaufort.
His paternal grandfather was Owain ap Meredudd ap Tudur, known as Owen Tudor (though he could easily have been called Owen Meredith, changing the name of a dynasty). Owen was a charismatic Welshman who fought at Agincourt in the service of Henry V. After Henry’s early death, Owen became the lover of the King’s young widow, Queen Catherine of Valois, mother to the infant Henry VI. They were married in secret, to the great consternation of the English nobility (Catherine was, to put it mildly, way out of Owen's league). They had at least two sons, Edmund and Jasper Tudor, half-brothers to the king. Catherine died at the age of 35 and in the absence of her protection Owen fell on rough times. He was imprisoned, escaped, spent some time on the run, was re-captured and again imprisoned. Henry VI finally pardoned him. Edmund and Jasper were officially recognized as half-brothers to the king and both became Earls, very much politically beholden to the the Lancastrian power in it’s conflict with the house of York. Edmund Tudor married Margaret Beaufort, so their son Henry’s descent from John of Gaunt make him marginally Lancastrian. At the end of the Wars of the Roses, he was the most prominent Lancastrian standing, despite the marginal nature of his claim.
Henry VII enhanced his position by marrying Elizabeth of York, daughter of Edward IV and granddaughter of Richard of York, who’s claims to the throne had launched the Wars of the Roses. This joined the Lancastrian and Yorkist lines under the Tudors as illustrated by the ‘Tudor Rose’ of both white and red. (The roses thing was really an afterthought, Lancasters and Yorks used a number of different emblems and only occasionally red and white roses.) So ‘Lancastrian’ is not really the right term for the Tudors after Henry VII, except in the context of comparison with other lines of descent. Dynastic names aside, Henry wore his Welshness proudly in naming his first son Arthur, after the hero of Welsh folk tales.