Pretty much every scene with John of Gaunt and Henry of Bolingbroke. It's pretty apparent how much Lancaster loves his son, even if he can't openly oppose Richard at all times. Special mention to Bolingbroke's departure, where Gaunt tries to lift his son's spirits in the face of his exile. It's also the last time they see each other.
In the 2013 RSC production with David Tennant as Richard — Richard comforting a weeping Aumerle at Flint Castle when they both know they're doomed. ("Aumerle, thou weep'st, my tender-hearted cousin!") Richard ditches his haughtiness and soothes him, holding him and kissing him tenderly; when it seems like Aumerle's recovering from his crying jag a little, Richard rather wryly sets his crown on Aumerle's head, only for this to instantly trigger another burst of crying and quickly putting it back on his own. Richard knows his time is running out, and his tenderness toward Aumerle (the last of his friends to survive and remain loyal) is honest, unaffected, and touching.
When they're both exiled, Bolingbroke asks Mowbray to just confess to killing Gloucester; he's never going to return to England, so why depart with a guilty conscience? Mowbray refuses because he didn't kill him. He gives his king one final farewell, and then leaves England forever.
In The Hollow Crown version, Queen Isabella begs for Richard to be exiled alongside her, it's obvious she cares deeply for him. Doubles as a Tear Jerker