delta-wye with dependent sources
If you delta-wye three components, and without mathematical errors, then the currents and voltages at the 3 nodes of the triangle you have transformed will be identical. This can allow you to series or parallel with other components to simplify the circuit more easily. Because you have changed the topology of the components within the triangle, their conditions will be very different.

Of course if you have made your target component vanish by doing a delta-wye, then you will have to get it back again at some stage to answer your question. If after a delta-wye, you absorb some of that component into others, then that's going to be tricky!

Where delta-wye might help is if you can use it to solve all the voltage and current sources. Then when you go back to the original problem, you know all the sources, and the solution of the original mesh might be easier.