There's a whole lot of talk about rising and falling in this play – so much so that it makes us feel a little seasick. When Henry forces Richard II to give up the crown, Shakespeare beats us over the head with the idea that the dramatic downfall of one king gives rise to another. Richard says as much as he literally hands over his crown:
Here, cousin, seize the crown;
On this side my hand, and on that side yours.
Now is this golden crown like a deep well
That owes two buckets, filling one another,
The emptier ever dancing in the air,
The other down, unseen and full of water:
That bucket down and full of tears am I,
Drinking my griefs, whilst you mount up on high. (4.1.2)
Here, Richard compares the crown to a well and says that he and Henry are like two water buckets on a pulley system (when one bucket goes down into the well, the other bucket rises up). Richard, of course, is like the bucket being lowered "down" into the well, filling with water (or tears). Henry, on the other hand, is like the emptier bucket that rises up, "dancing in the air," as the other is lowered down "unseen." Most of us don't get our drinking water from wells anymore, but we can still appreciate the astonishing image Richard creates with this metaphor. The point is also pretty clear: Bolingbroke's good luck is tied to Richard's misfortune. As Bolingbroke rises in power, Richard experiences a major downfall.
We also see this relationship between rising and falling in an earlier scene, when Bolingbroke shows up at Flint Castle and forces Richard to give up peacefully (3.3). When Bolingbroke arrives, Richard is standing up high on top of the castle walls. When Northumberland asks the king, "May it please you to come down?" (3.3.5), Richard makes a big deal out of the word "down" as he descends the steps to surrender:
Down, down I come; like glistering Phaethon,
Wanting the manage of unruly jades.
In the base court? Base court, where kings grow base,
To come at traitors' calls and do them grace.
In the base court? Come down? Down, court!
down, king! (3.3.5)
FYI, in Greek mythology, Phaethon was the guy who tried to drive his dad the sun god's chariot. When he couldn't control his horses, Zeus shot him out of the sky with a giant thunderbolt. By comparing his downfall to Phaethon's, Richard lets us know just how dramatic it is for him to lose the crown to Henry, even though this whole scene lacks the kind of violence we might expect when a king loses his power.
After Richard walks down the steps of Flint Castle, Bolingbroke greets him by kneeling down, presumably as a fake show of respect. Richard's not having any of this phony stuff. He says, "Up, cousin, up. Your heart is up, I know / Thus high at least, although your knee be low" (3.4.6). Translation: "Why are you down on your knee pretending that you're not taking my crown from me? You should be standing up to reflect your power and your elevated mood."