There is no transceiver/trigger that will have remote power control over the YN-560 or YN-560II. They don't have built-in RF with power control, and the single pin on the foot means the only signal that can be communicated over external radio triggers is sync.

You can remotely control the power on the YN-565EX if you use TTL-capable triggers, such as the YN-622s.

You basically have two choices, if you want to stay with Yongnuo:

• Dump all your units, and go with the YN-560-TX triggering system. This is manual-only triggering (no TTL or HSS), but you will have remote power control if you use a YN-560-TX or a YN-560IV or YN-660 on the camera hotshoe. As slaves, you can use the YN-560III, YN-560IV, or YN-660. You cannot add Yongnuo TTL gear into this setup without losing TTL capabilities to make it compatible with the triggering system. Any non-Yongnuo speedlights cannot be power-controlled through RF-603II or RF-605 transceivers. Yongnuo's YN-622 and RF-60x systems are incompatible.

• Dump the -560 units, and get TTL-capable speedlights and add YN-622 transceivers, or get the YN-685EX (which has a built in 622 receiver). You can use a 622 or 622-TX unit as your on-camera transmitter. But any flashes you want to add into this setup must be TTL-capable (and if you're a Canon shooter, menu-commandable) if you want to set manual power levels (vs. TTL ratios).

There's also a third option that may make more sense for future growth (well at least this month; low-end Chinese flash gear churns at an alarming rate. The FlashHavoc blog is a good way to keep up with what's current):

• Dump all your units and go with the Godox X system. The UI/UX isn't quite as nice for Canon (especially if you're shooting with a pre-2012 body), but you can easily mix TTL and manual-only speedlights, and, unlike the RF-60x/YN-560 system, you will have HSS with either (assuming you're shooting with a body that can do FP/HSS--Nikon 3x00 and 5x00 bodies can't). The TT600--the Godox equivalent of the YN-560III/IV/-660 plays nicely in the same system with the TT685 (the Godox equivalent of the YN-685--only it has a transceiver built in, not just a receiver). And the same radio system is built into Godox's other lights: the Li-Ion powered speedlights (V850II/V860II), the bare bulb flashes (AD200, AD360II), and the monolights (SKII series, QSII series, QT600II, and AD600). If you ever want to go bigger than speedlights, Godox gives you a lot of choices; Yongnuo not so much.

See also: What are the Yongnuo flash naming conventions?