My understanding is that Lutherans have some doctrine of sovereign election/predestination, yet they also affirm unlimited atonement and that some do go on to shipwreck their faith. What are the mechanics of how these doctrines interplay in confessional Lutheran theology?
Lutherans believe that God elects to salvation, but not to damnation. That is to say, we believe that our salvation is due to God's election on account of His grace and mercy for the sake of Christ (i.e. God chooses us because He is merciful, not because we've done something to earn it or because He sees that we will do something to merit salvation). See, for example, Ephesians 2:1-10 and Romans 5:6-21.
By way of comparison, Calvinists would then typically go on to say that the logical corollary is that God therefore chooses who is damned; i.e. He "elects to damnation." However, Lutherans say that this is not correct. We believe that Christ died for the sins of all people (e.g. 1 John 2:2) and that God wants all people to be saved (e.g. 1 Timothy 2:4).
Lutherans also believe that it is God Himself who brings people to faith through the working of the Holy Spirit through the Gospel in its spoken, written, and sacramental forms (see, for example Romans 10:10-17 where Paul talks about faith coming from hearing through the "Word of Christ"). So, even our very faith is a work of God (a point also touched on in the earlier example from Ephesians 2:8-9).
To the question then, of why some people are not saved, even though Christ died for their sins and even though God wants them to be saved, Lutherans would answer, "We don't know." It's a paradox that we don't believe we can resolve (Paul seems to touch on this also in Romans 9).
We often make a distinction between the "revealed God" and the "hidden God" that's useful in considering this paradox. The "revealed God" is the aspect of God that He reveals to us in the Scriptures; i.e. He created all things, He came in the flesh to redeem all things, Christ died for all people, He wants all people to be saved. The "hidden God" is the aspect of God that He does not reveal to us; it's His hidden will that He does not show us. Lutherans believe that questions like why some people are saved and others are not and why good people sometimes suffer and bad people sometimes prosper fall into this category of the "hidden God." We can't know or discover the answers through our own efforts, because God hasn't told us the answers. It's similar to how young children only know their father based on what he has told them about himself.
In the end, then, we lean back on what God has clearly revealed to us and go from there. So, we can confidently tell people that Christ died for their sins and that the Gospel is for them, because this is what God tells us in the Scriptures. Then, we trust in God to bring them to faith in the Gospel for their salvation. Maybe another way of saying it is that we do all we can in spreading the Gospel, and trust in God to work through this proclamation to save people by bringing them to faith.
We also believe that people can fall from faith (e.g. Luke 8:13, Hebrews 6:1-8). Typically, we believe this happens when people persist in some sort of unrepentant sin after coming to faith. That is, they fight against the working of the Holy Spirit in them, until they ultimately destroy their faith. For this reason, Lutherans talk about a daily "living out of our baptism" by dying and rising each day through repentance and forgiveness. Our entire lives are ones of confession and absolution, whereby we confess our sins (e.g. in prayer or to the pastor) and receive absolution (e.g. in the Gospel and Sacraments, or the pronouncement from the pastor). Note, though, that God's forgiveness is already there for us before we confess and before we repent; confession and repentance, in essence, open our eyes to the fact that we need God's forgiveness and mercy.