The band connected sum and the second Kirby move for higher-dimensional links

Let f:Sq Sq→ Sᵐ be a link (i.e. an embedding). How does (the isotopy class of) the knot Sq→ Sᵐ obtained by embedded connected sum of the components of f depend on f? Define a link σ f:Sq Sq→ Sᵐ as follows. The first component of σ f is the `standardly shifted' first component of f. The second component of σ f is the embedded connected sum of the components of f. How does (the isotopy class of) σ f depend on f? How does (the isotopy class of) the link Sq Sq→ Sᵐ obtained by embedded connected sum of the last two components of a link g:Sq₁ Sq₂ Sq₃→ Sᵐ depend on g? We give the answers for the `first non-trivial case' $q=4k-1$ and $m=6k$. The first answer was used by S. Avvakumov for classification of linked 3-manifolds in S⁶.