There are a couple of Targ locations which create an effect for the phase but refer to challenge singularly - this has caused some confusion

(the Dragon must be participating in a challenge in order to kneel the location)

Kneel Astapor to choose 1 participating Dragon character you control. Until the end of the phase, that character gets +2 STR. If you win the challenge, stand that character.

Kneel Mereen to choose 1 participating Dragon character you control. Until the end of the phase, that character gains "Immune to opponents' trigggered effects." If you win the challenge draw 1 card for each participating Dragon character you control

We assume that the last part refers to only the current challenge - is this correct?

Correct. If you win the challenge during which the effect was triggered, you stand the character for Astapor resp. draw the card(s) for Mereen. The STR buff and the gained immunity, however, last for the duration of the phase.