What is the process for remarried couples to convert to Catholicism? I was reading and trying to understand Amoris Laetitia 
  and came up with this question below.

The documents talk about irregular couples and basically say that everyone undergoes different timing of conversion.

This is a situation:
A non-practicing Catholic couple "divorces" and re-marry by state. They end up having families, let's say with 3 or more kids with another husband/wife. According to the Catholic teaching, they live in adultery. Now, they start to be interested in spiritual life and undergo what we call a conversion. But they are stuck because they can't just split and leave their kids or whatever dependencies they have. 

I know the ideal would be to get an annulment and solve this problem. But let's say they aren't entitled to an annulment. 
My question is what does the church offer to such couples as a solution?
Suggesting to live like a brother and sister isn't going to work unless they stop living in the same household but that might not be ideal for kids (I guess) or it can still cause a scandal.

Is splitting couples really the only way to go if they want to live a sacramental life? I know this happened to St. Augustine but his situation wasn't that complicated. Do we have other examples from the past where families needed to separate in order to return back to living in the state of grace?

Have Catholic theologians written on this subject and if so could someone explain it to me?
G.K. Chesterton discusses the logical absurdity of pretending to &quot;cancel&quot; a life long vow with a piece of paper from a government in &quot;The Superstition of Divorce&quot;.  I highly recommend reading or listening to (free on Librivox) this short book.
Disclaimer, I am not Catholic, but try to study their positions seriously.  I think a friendly outside perspective may be helpful.
Catholics focus on verifiable objective facts.  The couple did make vows as nominal Catholics according to the premises of the question.  This can be verified from Church records, signed and witnessed.  Protestants focus on subjective realities.  Possibly, the individuals were just following family tradition getting married in a Catholic Church, and did not comprehend the seriousness of their vow.  The Catholic annulment process attempts to address the situation of making vows in ignorance or under compulsion (forced marriage) or false pretences.
If the the original vows were real, then both parties have been living in adultery, and had multiple kids in adultery.  The kids are going to be scarred.  The Law of Moses said: &quot;No one of illegitimate (adulterous) birth shall enter the assembly of the LORD unto the 10th generation.&quot;
Nevertheless, Children of &quot;putative marriages&quot; (Canon 1061.3 An invalid marriage is said to be putative if it has been celebrated in good faith by at least one party. It ceases to be such when both parties become certain of its nullity.) are considered legitimate by the Catholic Church.
There is no way to unscramble this egg.  The Catholic prescription of &quot;living as brother and sister&quot; with the adulterous partners is as good as anything a Protestant denomination might suggest.  The Catholic Catechism paragraph 1649 and 1650 ends with:

&quot;Reconciliation through the sacrament of Penance can be granted only to those who have repented for having violated the sign of the covenant and of fidelity to Christ, and who are committed to living in complete continence.&quot;
Toward Christians who live in this situation, and who often keep the faith and desire to bring up their children in a Christian manner, priests and the whole community must manifest an attentive solicitude, so that they do not consider themselves separated from the Church, in whose life they can and must participate as baptized persons:
They should be encouraged to listen to the Word of God, to attend the Sacrifice of the Mass, to persevere in prayer, to contribute to works of charity and to community efforts for justice, to bring up their children in the Christian faith, to cultivate the spirit and practice of penance and thus implore, day by day, God's grace.

If the original vows were made in ignorance by both parties, then we have two (nominally Catholic) pagan families, at least two members of which now seek to obey Christ and rejoin the Catholic Church.  In this case, the original marriage was effectively pagan fornication.  This statement does not dismiss the Catholic ceremony - the couple disrespected it while making serious vows they did not mean.  In this case, the way forward is clear - stay with the current partners.  If the Catholic church cannot annul the vows made in ignorance (due to their unarguable objective reality), then the new marriages cannot be validated with (real this time) vows by the Church, and &quot;living as brother and sister&quot; is again the only option to avoid fornication in the eyes of the Church.
Since the first answer got voted down, I'm sure this one will too.  Broken eggs make a big mess.  GOD HATES DIVORCE.  Malachi 2:13-16
Not mentioned in the question is whether the new partners are also Believers seeking to obey Christ.  Are they also Catholic?  Or something else?
Now, because you are seeking to Obey Christ, it is up to Christ to make something beautiful out of this screwed up mess.  You don't have to understand the whole picture, or know the ultimate outcome.  Just do what is clear, and trust God to work all things together for good.
Addendum from comments:
Canon law site: http://www.catholicdoors.com/misc/marriage/canonlaw.htm
This confirms that the first marriage is the only valid one - but doesn't say what to do about this mess.  If there were no children, the clear course would be to separate from the adulterous partners.
It also has a web form to ask questions of a presumably knowledgeable Catholic.  I had previously discussed a similar situation with a serious and knowledgeable Catholic friend - not a source I can link to, but that is where my understanding about &quot;living as brother and sister&quot; as the Catholic approach to dealing with the mess comes from.  The Canon law page linked above confirms the necessity of protecting children in any decision.
The confessions (e.g. Westminster Confession) of most Protestant groups are just as strict about divorce - there are different approaches to dealing with the messes caused by divorce and remarriage.  The practice of these ideals has deteriorated horribly in all communions, including Catholic.
We all need to repent - Holy Marriage is the divinely chosen living illustration of Christ and the Church. 2 Corinthians 11:2 Revelation 19:7-9 When marriage is defiled, Christ is obscured.