Is there any way to mapping point between 2 elliptic curves? Consider the elliptic curve  $E_1: y^2 = x^3+7$ over $\mathbb F_{17}$ with the base point $G=(15, 13)$
and the second elliptic curve $E_2: y^2 = x^3+7$ over $\mathbb F_{31}$ with the same base point $G=(15, 13)$.
My question is: is there any way to calculate the equivalent point of $\mathbb F_{31}$ based on $\mathbb F_{17}$?
For example: with $7G = (10, 15)$ of curve $\mathbb F_{17}$, how to calculate $7G$ of $\mathbb F_{31}$? The result should be $7G = (12, 14)$ on $\mathbb F_{31}$.
Below are all points of the two curves:
#----Curve F17-------#
  1G       2G       3G       4G       5G       6G       7G       8G       9G      10G      11G      12G      13G      14G      15G      16G     17G
(15,13)  ( 2,10)  ( 8, 3)  (12, 1)  ( 6, 6)  ( 5, 8)  (10,15)  ( 1,12)  ( 3, 0)  ( 1, 5)  (10, 2)  ( 5, 9)  ( 6,11)  (12,16)  ( 8,14)  ( 2, 7)  (15, 4)

#----Curve F31-------#
  1G       2G       3G       4G       5G       6G       7G       8G       9G      10G      11G      12G      13G      14G      15G      16G      17G      18G      19G      20G      21G      22G      23G      24G      25G      26G      27G      28G      29G      30G      31G
(15,13)  (29,17)  ( 1,22)  (20,19)  (21,17)  (23,23)  (12,14)  (11,27)  (25,22)  ( 7,19)  (27,27)  ( 5, 9)  ( 0,24)  ( 4,12)  (22,23)  ( 3,13)  (13,18)  (17,23)  (24, 4)  (24,27)  (17, 8)  (13,13)  ( 3,18)  (22, 8)  ( 4,19)  ( 0, 7)  ( 5,22)  (27, 4)  ( 7,12)  (25, 9)  (11, 4)
I wrote this since it is too long to comment;
First the SageMath code
K =  GF(43)
E = EllipticCurve(K,[0,7])
print(E)
print(E.order())
print(E.abelian_group())

G = E(21,25)

With $K$ one can use different fields, the $[a=0,b=7]$ is for $y^2 = x^3 + ax + b$.
This codes give us a lot of information about the curve

print(E) : The curve information:
Elliptic Curve defined by y^2 = x^3 + 7 over Finite Field of size 43

E.order() : Print the number of the rational points on the curve:
31

E.abelian_group(): The group structure of the curve:
Additive abelian group isomorphic to Z/31 embedded in Abelian group of points on Elliptic Curve defined by y^2 = x^3 + 7 over Finite Field of size 43


The last one is important since to exactly map a point $g$ and its multiples, they must have the same order in each group.
Actually, to map a curve to another, or a subgroup, one doesn't need the same $G$ to happen in both. The order is enough. We have to look for orders of the elements.
Let's look at your curves;

EllipticCurve(GF(17),[0,7])

Elliptic Curve defined by y^2 = x^3 + 7 over Finite Field of size 17
18
Additive abelian group isomorphic to Z/18 embedded in Abelian group of points on Elliptic Curve defined by y^2 = x^3 + 7 over Finite Field of size 17


E = EllipticCurve(GF(31),[0,7])

Elliptic Curve defined by y^2 = x^3 + 7 over Finite Field of size 31
21
Additive abelian group isomorphic to Z/21 embedded in Abelian group of points on Elliptic Curve defined by y^2 = x^3 + 7 over Finite Field of size 31


E = EllipticCurve(GF(43),[0,7])

Elliptic Curve defined by y^2 = x^3 + 7 over Finite Field of size 43
31
Additive abelian group isomorphic to Z/31 embedded in Abelian group of points on Elliptic Curve defined by y^2 = x^3 + 7 over Finite Field of size 43


E = EllipticCurve(GF(67),[0,7])

Elliptic Curve defined by y^2 = x^3 + 7 over Finite Field of size 67
79
Additive abelian group isomorphic to Z/79 embedded in Abelian group of points on Elliptic Curve defined by y^2 = x^3 + 7 over Finite Field of size 67



They are all isomorphic to $Z/18,Z/21,Z/31,$and $Z/79$, respectively.
The orders;
for i in (1..E.order()):
    print(E[i], E[i].order())


In the first curve, the orders of the elements are $1,3,2,6,9,18$
In the second curve, the orders of the elements are $1,3,7,21$

So, we have common order of $3$, then we can map these elements (5,8) and $(0,10)$ to each other.

A sample curve that is not isomorphic Z/n
E = EllipticCurve(GF(25, 'x'), [1, 1])

Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field in z2 of size 5^2
27
Additive abelian group isomorphic to Z/9 + Z/3 embedded in Abelian group of points on Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field in z2 of size 5^2


Birational Equivalent is another story!