Yes, movement between tracks is possible. An Advanced Algebra student who excels as a freshman would be recommended to take an honors math class as a sophomore. An exceptional student in regular Algebra I could also move into the honors track.
Honors College Algebra uses a college algebra text and assumes a strong algebra foundation (based on the entrance exam and the optional math placement test). Most of these students will already have had two years of algebra and possibly a geometry class.
Algebra and Advanced Algebra use the same text but the Advanced class covers more concepts in more depth and at a faster pace. Students in the Advanced Algebra classes are also better prepared to move up to honors math as sophomores.