In the Beatitudes, Jesus says:Yes. The Beatitudes is not a pick and choose list. You can't say, "I want to inherit the earth so I'm going to be meek, but righteousness isn't very high on my list of priorities so I'm not going after that one right now." Or, "Righteousness just seems so unattainable that I'm not going to even try for that one."
(ESV) Matthew 5:6 "Blessed are those who hunger and thirst for righteousness, for they shall be satisfied."
Is this a characteristic that will be found in all Christians?
The Beatitudes are not a unit unto themselves either. They are part of Christ's larger Sermon on the Mount. The Sermon on the Mount is Christ's teaching on Kingdom living. Since we believe that the Kingdom is present now, all those who are part of the Kingdom are bound by it.
Granted, these Beatitudes are not in the imperative. They are not commands. But they are descriptions of Christians and the rewards that await them. But I would also say that absence of these rewards equals absence of salvation. For instance, the pure in heart will see God. We know that all Christians will see God. Therefore all Christians will be pure in heart. This doesn't mean that I'm perfect now. I still have the rest of my life to live and grow in sanctification, however long or short that may be. But I do long for the day when sin is done away with in me and I am truly pure in heart and receive my reward, seeing and knowing God even as I see and know my family.
So I would say that longing for righteousness is a characteristic of all true believers. They are in the process of being filled with righteousness now, and will one day be righteous.