A graph \(X\) is said to be shuffled by a subgroup \(G\) of its automorphism group \(\Aut(X)\) if for every infinite \(C \subseteq V(X)\) with finite boundary \(\partial C\) and every finite \(F \subseteq V(X)\), there exists \(\sigma \in G\) such that \(\sigma (F) \subseteq C\). The first main result is that every connected graph \(X\) of finite diameter is shuffled by any \(G \leq \Aut(X)\) that acts transitively on \(V(X)\). One may replace ``of finite diameter'' with ``having more than one end''. A bisection of \(X\) is a partition \(\{C_ 1,F,C_ 2\}\) of \(V(X)\), where \(C_ 1\) and \(C_ 2\) are infinite, \(F\) is finite, and \(\partial C_ 1\), \(\partial C_ 2 \subseteq F\). The second main result is that if \(G\) acts transitively on \(V(X)\), then for any bisection \(\{C_ 1,F,C_ 2\}\) of \(V(X)\), there exists \(\sigma \in G\) with no finite orbit such that \(\sigma\) and \(\sigma^{-1}\) have distinct directions and \(\sigma(F \cup C_ 1) \subseteq C_ 1\). Several results that are known for locally finite graphs are then proved with the local finiteness restriction removed. Among these are: (1) Every transitive connected graph with \(>2\) ends has no free end; (2) Every transitive connected graph has 1,2 or \(\geq 2^{\aleph_ 0}\) ends; (3) In a transitive connected graph with \(\geq 2\) ends, an automorphism is bounded if and only if it fixes every end.