When you use “Union Rider”’s effect to equip a Union Monster to it, because the Union Monster was not equipped to “Union Rider” by the Union Monster’s effect, the bonuses of the Union Monster are not applied, including the “If the monster that this card is equipped to is destroyed in battle, this card is destroyed instead.” effect.
You can only use the effect of “Union Rider” during your Main Phase 1 or 2.
If you use “Combination Attack” or “Formation Union” on your “Union Rider” equipped with your opponent’s Union Monster, the Union Monster is Special Summoned onto your (the controller of “Union Rider”’s) side of the field.
If you equip your “Union Rider” with your opponent’s Union Monster, and attack with “Union Rider”, then use “Combination Attack”, you will then control both the “Union Rider” and the Union Monster on your side of the field, and both may attack (again) that turn.
The Union Monster you equip to your “Union Rider” is actually placed in your Spell & Trap Card Zone, so it counts for you for “Mage Power”, etc.
You must have a Spell & Trap Card Zone space vacant to activate your “Union Rider”’s effect.