Can a player use Relentless Pursuit to move a ready unit to a battlefield, then conquer, then return to base still ready, then move it to the other battlefield, then conquer, and return to base?

Yes, you can perform that sequence. According to RiftJudge FAQ #8229 and RiftJudge FAQ #9079, the movement granted by Relentless Pursuit is a special effect that does not consume the unit's "ready" status, nor does it count as a "Standard Move."

Here is the breakdown of how this works:

The Sequence

1. First Battlefield: You play Relentless Pursuit to move your ready unit to the first battlefield.
2. Conquer: The unit conquers the battlefield. This triggers the "When I conquer, you may move me to my base" ability granted by the spell.
3. Return to Base: You resolve the trigger to move the unit back to your base. Because this is an effect of the spell and not a "Standard Move," the unit remains ready.
4. Second Battlefield: Since the unit is still ready, you can perform a Standard Move (Rule 144) to move it from your base to the second battlefield.
5. Conquer Again: The unit conquers the second battlefield. This triggers the "When I conquer, you may move me to my base" ability again.
6. Return to Base: You resolve the trigger to move the unit back to your base.

Key Rules & Clarifications

• Ready Status: Moving via the effect of Relentless Pursuit does not exhaust the unit. As long as the unit was ready before the move, it remains ready after the move.
• Standard Move vs. Ability: The move to base is an effect of the spell, not a "Standard Move." Therefore, it does not consume the unit's ability to perform a Standard Move later in the turn.
• Conquering Multiple Battlefields: While you can move the unit to multiple battlefields, remember that you can only score 1 point per battlefield per turn (RiftJudge FAQ #7610). If you have already scored the first battlefield this turn, you will not gain a second point for conquering it again, though you will still gain a point for conquering the second, un-scored battlefield.
• Simultaneous Triggers: If your unit has its own printed "ready" ability (like Lucian's), it will trigger simultaneously with the return-to-base ability granted by Relentless Pursuit. You may choose the order in which to resolve them (RiftJudge FAQ #8229).