Does the growth rate of a neoclassical production function converge as all input factors grow with constant, but different growth rates?
I am not sure that elasticities converge for arbitrary growth rates.  

Here is a treatment of the issue. Subscripts in $F$ denote partial derivatives. I will omit the time subscript. By definition, the elasticity of the $i$-th input is the ratio of marginal product ($F_i$) over Average Product $(F/x_i)$. So for an elasticity to stabilize intertemporally, it must be the case that the growth rate of the marginal product  is equal to (or equalizes a the limit with) the growth rate of the average product.  

For input $x_{1}$ we have

$$\frac {d}{dt} F_1 = F_{11}\dot x_{1} +F_{12}\dot x_{2} +...+F_{1n}\dot x_{n}$$

All inputs are strictly positive so we can multiply and divide, and also divide by $F_1$ to obtain the growth rate (denoted by a tilde)

$$\tilde {F_1} = \frac {dF_1/dt}{F_1}  = \frac{F_{11} x_1}{F_1}\frac{\dot x_{1}}{x_1} + ... +\frac{F_{1n} x_n}{F_1}\frac{\dot x_{n}}{x_n}$$

The first ratio in each term is the elasticity of the $F_1$ with respect to each input, say $\eta_{1k}$. So, 

$$\tilde {F_1} = \sum_{k=1}^n\eta_{1k}g_k $$

and analogously for the other inputs.

Turning to the average product of the first input, we have

$$\frac {d}{dt} (F/x_1 ) = \frac {F_1x_1 - F}{x_1^2}\dot x_1 + \frac {F_2}{x_1}\dot x_2+...+\frac {F_n}{x_1}\dot x_n$$

Manipulating, 

$$\frac {d}{dt} (F/x_1 ) = \left[F_1-\frac {F}{x_1} \right]g_1 + \frac{F}{x_1}\frac{F_2x_2}{F} \frac {\dot x_2}{x_2}+...+\frac{F}{x_1}\frac{F_nx_n}{F} \frac {\dot x_n}{x_n}$$

$$= \frac {F}{x_1}\cdot \Big( (e_1-1)g_1 + e_2g_2 + ...+ e_ng_n\Big)$$

$$\implies \frac {d(F/x_1)/dt}{(F/x)} = \tilde F - g_1 = \sum_{k=1}^ne_kg_k - g_1$$

So for all elasticities to stabilize (and so for $\tilde F$ to stabilize) we want, at least eventually, that

$$ \forall i=1,...,n\;\;\;  \sum_{k=1}^n\eta_{ik}g_k = \sum_{k=1}^ne_kg_k - g_i$$

Denoting $\mathbf H$ the $n\times n$ the matrix of the elasticities of the marginal products, $\mathbf I_n$ the identity matrix, $\mathbf e$  an $n \times 1$ vector containing the output elasticities $e_i$, $\mathbf i$ a column vector of ones, and $\mathbf g$ the $n \times 1$ column vector of growth rates of inputs, we can write the conditions for all elasticities to be stable compactly , as

$$\big(\mathbf H + \mathbf I_n - (\mathbf i \oplus \mathbf e')\big)\mathbf g =0$$

I note that the sum-of-matrices term does not depend at all on the growth rates vector. So the above orthogonality condition cannot hold for arbitrary $\mathbf g$ (except if I am neglecting some property stemming from the homogeneity of $F$).