Morphing between two functions

You might be interested in the approach with optimal transport.

Let $F(x)=\int_{-\infty}^x f(y)\,dy$ and $G(x)=\int_{-\infty}^x g(y)\,dy$ be the repartition functions. Then $T(x)=G^{-1}\bigl(F(x)\bigr)$ is the optimal transport map from $f$ to $g$. The map $T_t(x)=(1-t)x+tT(x)$ is the displacement geodesic, so that the intermediate densities are given by $(T_t)_\#f$.

f[x_] := Exp[-(x + 3)^2]
g[x_] := 1/2 Exp[-(x - 3)^2/4]
F[x_] = Integrate[f[x], {x, -\[Infinity], x}];
G[x_] = Integrate[g[x], {x, -\[Infinity], x}];
Ginv[q_] = InverseFunction[G][q];
T[t_, x_] = (1 - t) x + t Ginv[F[x]] // Simplify;
dT[t_, x_] = D[T[t, x], x] // Simplify;
ParametricPlot[Evaluate@Table[
   {T[t, x], f[x]/dT[t, x]}, {t, 0, 1, .1}],
   {x, -10, 5}, PlotRange -> All, AspectRatio -> 1/2]

In the case of Gaussians, as in your example, the interpolation is still Gaussian, and the explicit formula can be found here or here or here...

Clear["Global`*"]

f[x_] := Exp[-(x + 3)^2]
g[x_] := 1/2 Exp[-(x - 3)^2/4]

Treating f and g as unnormalized distributions

distf = ProbabilityDistribution[f[x],
   {x, -Infinity, Infinity}, Method -> "Normalize"];

distg = ProbabilityDistribution[g[x],
   {x, -Infinity, Infinity}, Method -> "Normalize"];

disth = TransformedDistribution[(x + y)/2,
   {x \[Distributed] distf, y \[Distributed] distg}];

data = RandomVariate[disth, 1000];

h[x_] = Integrate[f[x] + g[x], {x, -Infinity, Infinity}]*
  PDF[EstimatedDistribution[data,
    NormalDistribution[m, s]], x]

(* 1.81073 E^(-0.819682 (0.0440864 + x)^2) *)

Plot[{f[x], g[x], h[x]}, {x, -10, 10},
 PlotRange -> All,
 PlotLegends -> Placed["Expressions", {.75, .6}]]

EDIT: Or for a zero mean

h[x_] = Integrate[f[x] + g[x], {x, -Infinity, Infinity}]*
  PDF[EstimatedDistribution[data,
    NormalDistribution[0, s]], x]

(* 1.81039 E^(-0.819382 x^2) *)

Plot[{f[x], g[x], h[x]}, {x, -10, 10},
 PlotRange -> All,
 PlotLegends -> Placed["Expressions", {.75, .6}]]