Movement in the inner tier is one space a turn. But do you have to stop your movement on the last space in the middle tier and go into inner tier next turn? I could not find and info on this.

You do not have to stop there and move on next turn. The space says "If you have a Relic, you may spend all remaining movement points to move to the Warp Rift".

So, as long as you land there with at least one movement point left, you may proceed to the Rift. This is covered on page 21 of the rules under "Movement Boxes".

TK