Charcoal, 19 bytes

ＡθＡηＥ§θ⁰§§θＬΦ⊕κ№ηλκ

Try it online! Link is to verbose version of code. Takes input as a pair of string gene sequences and a 0-indexed list of crossing points. Explanation:

Ａθ                  Input the pair of gene sequences into `q`
  Ａη                Input the list of crossing points into `h`
    Ｅ§θ⁰            Loop over one of the gene sequences
              κ     Current index
             ⊕      Incremented
            Φ  №ηλ  Intersect implicit range with crossing points
           Ｌ        Take the length
         §θ         Cyclically index into the pair of gene sequences
        §         κ Take the appropriate element of that sequence
                    Implicitly output on separate lines

Charcoal, 19 bytes

ＡθＡηＥ§θ⁰§§θＬΦ⊕κ№ηλκ

Try it online! Link is to verbose version of code. Takes input as a pair of string gene sequences and a 0-indexed list of crossing points. Explanation:

Ａθ                  Input the pair of gene sequences into `q`
  Ａη                Input the list of crossing points into `h`
    Ｅ§θ⁰            Loop over one of the gene sequences
              κ     Current index
             ⊕      Incremented
            Φ  №ηλ  Intersect implicit range with crossing points
           Ｌ        Take the length
         §θ         Cyclically index into the pair of gene sequences
        §         κ Take the appropriate element of that sequence
                    Implicitly output on separate lines

Alternatively, ⭆ could be subsituted for Ｅ to print the result as a string. Try it online!