To become weary can mean to be in danger of losing heart. There are not many things more scary for a Christian than to end up losing heart. I then can find myself in a kind of stupor in which I think nothing I do matters much. Well, it's all of grace to be sure. But this grace lifts us to be a part of what God in Christ is doing in the world. To have our part together with others in Jesus.
Losing heart can be spread to others. But taking heart by grace can likewise help others do the same. I think this is conveyed to each other in our spirits by the Spirit, and it's through what in Jesus we are and do. But above all, it's in answer to prayer. We actually may feel quite weak and even struggling, but in answer to prayer God can give us "heart" to keep on doing good works and his will.
I think as we go on in Jesus in this life, we can learn more and more to take heart no matter what. Of course we can't retreat into a life of sin or indulging the flesh, in other words doing what we feel like doing, and then expect to find heart in God. Taking heart in God through Christ is only for us as we seek together, to follow the way of Christ. At times left by ourselves to do so, but always wanting to take someone along with us.