I use the word "problems" in quotes, because the first two of these are usually only problems to those who do not believe in Baptist succession, and the third is a self-created problem by those who do.
Definition. The idea of church perpetuity or successionism may vary a bit among Baptists, but in this post I am using it to mean: "there has never been a day since the organization of the first New Testament church in which there was no genuine church of the New Testament faith existing on earth, and that this will continue to the end of the age."
Historical problem. The historical problem is a problem of the preservation of valid records and the interpretation of those records. No local church can demonstrate a valid succession all the way through history back to Christ. Neither is it historically demonstrable that every day and every year has had a New Testament church existing. This seems to be a problem for some who reject perpetuity, but those whose accept it usually do so on the basis of the words of Jesus "the gates of hell shall not prevail against it", et al. Church perpetuity is based on the promise of Jesus Christ, and historical research is just a sideline. Historical research is much more likely to run across some heresy that puts one "out of the line" of the true church than to discover the links in a chain all the way back to the time of Jesus.
Theological problem. The theological problem exists for those who believe the promise of continuance is only for some kind of universal mystical invisible thing they call a church. This creates a debate over the meaning of scripture between those who believe that and those who believe in the promise for a visible church.
Historical/theological problem. This last one is my main point. There is a historical/theological problem for those Landmarkers who build up complicated theories on what makes a church lose its candlestick, when in fact they have churches of that very sort in their theological lineage. Can a church that teaches the universal church teaching be a New Testament church? Can a church that practices close communion be a New Testament church? Can a church that practices open communion be a New Testament church? Etc. This is where some create a theological/historical problem. Their theology does not match their history. If a gathering of believers holding universal church and/or close communion is not a Scriptural church, how does one deal with such in their church ancestry? Let me say plainly that I have seen time and again folks boldly proclaim this type church or that type church is not scriptural, to only stick their collective heads in the sand when challenged with such a church in their own ancestry.
Most successionists probably cannot trace their Baptist heritage back more than one hundred years without finding close communion, and not more than 300 without finding some kind of church of the redeemed idea. If so, one might need to check out his or her own baptism rather than just pointing fingers at others.