This paper studies some properties of (n,m)-homogeneous isotopies of topological groupoids with (n,m)-identities. The concept of (n,m)-identities was introduced by M.M. Choban and L.L. Chiriac. We extend some statements from the theory of topological groups to the class of topological (n,m)-homogeneous quasigroups. It is shown that if P is an open compact neighborhood of an (1,2)-identity e of an AG-quasigroups G, then P contains an open compact AG-subquasigroup Q with (1,2)-identity e. A new method of constructing non-associative medial and paramedial topological quasigroups is given. We study properties of AG-topological quasigroups and paramedial quasigroups with multiple identities.