Let an elliptic curve $E$, and 2 points on such curve $P$ and $O$ the methods I’m talking about consist in creating a weaker elliptic curve $F$ and mapping $P$ and $O$ to $F$ while successfully preserving the discrete logarithm $s$ between the two such as $F=s×O$.

The most well know example is moving elliptic curves into hyperelliptic curves but as far I’m aware this only work on extension fields of medium sized degree (and not over prime field). But of course, I’m more interested in curve over prime fields.
As an example outside hyperelliptic curve, is there a case that consider building a different curve defined on a field from different characteristic while preserving the same order in a subgroup ?