How should the term “his cross” and “the cross” in Mat. 16:24, Mrk. 8:34, 10:21, and Lu. 9:23 be contextually understood?

In each of those verses, the people hearing the Lord would understand that he was speaking of their willingness to die, give up their life.  The context is the tribulation in which their life will be on the line for following Jesus as the Christ.  The Holy Spirit which the believing remnant will be empowered with, will be the means they can endure to the end by life or death, and afterwards enter into the Kingdom.