When we have a vector bundle ( of finite rank ) over a scheme, that induces a flat pull back map of cycle classes, which can be shown to be isomorphism.
The surjectivity holds for affine bundle as well, this can be reduced to the case where the bundle is trivial using Noetherian induction : the cycle classes of a scheme can be “roughly” divided into the classes in an open subset and those in the complement ( this can be described by an exact sequence, which I would name the structural exact sequence ).
The injectivity is much harder to show, we need extra structures. First we need to complete the bundle projectively, with its projectivization added as the hyperplane at infinity. Then using Noetherian induction, we can show that there is an “O(1) decomposition” of each cycle class in the projective bundle as a ordered sum of classes : each of which is represented by the iterated action of the canonical line bundle on a pull back class. This fits nicely with the description of a projective space recursively as piecing an affine space with a projective space of 1 dimension less at infinity. Then the injectivity comes as a corollary of these added structure, using the “structural exact sequence” on the bundles, which are all compatible with the projection onto the base scheme.
There is a more descriptive description of the Gysin homomorphism using the the universal quotient bundle on the projective completion . To get the class on the base scheme via the Gysin morphism, first we take the closure of the class inside the projective completion of the bundle, intersect it with the highest Chern class of the universal quotient bundle, and the project it onto the base scheme.