Under what circumstances would a non-Catholic marriage be considered invalid by the Catholic Church?

I am of the understanding that generally, the Catholic Church recognizes non-Catholic marriages as valid marriages. So in general, if a non-Catholic couple converts to Catholicism, they will not have to re-certify their marriage (or even if just one of the two becomes Catholic). I'm wondering under what circumstances one might not be able to transfer his/her marriage in this way, i.e. when would a converting person's marriage *not* be considered valid by the Church. For instance (and these things may be common in some cultures): * The person could be a remarried divorcee, * The couple may be first cousins, * One or both may have been underage at the time of the marriage, or still underage at the present time, * The marriage may have been conducted without consent of one or both parties, * The person may be a polygamist. Which situations which would definitely never be considered a valid marriage? If some are considered on a case-by-case basis, how are cases determined?