Let's assume that the k-variety X is irreducible and projective. The
latter assumption is equivalent to that there is an ample line bundle L
on X. Now consider the sequence F(n) = F (x) L^{(x)n} of the n-th twist
of the coherent O_X-module F w.r.t. L (n>=0) ["(x)" denotes the tensor
product over the structure sheaf O_X of X].

Then is is known that

        \chi(F)(n) := dim_k \Gamma(X,F(n))

is a polynomial in n (for n>>0). Since Supp(F)=X, the degree  d  of
\chi(F) is equal to dim(X) and the leading coefficient of f_F is a the
form rk(F).deg(L)/(d!).

Here rk(F) denotes the rank of F, i.e. the dimension of the stalk of F
in the generic point of X. Note that rk(F(x)G) = rk(F).rk(G) . If F is
locally free - as you do -, rk(F) is simply the rank of F in every point
of X. deg(L) is a positive integer associated to the line bundle L,
called the degree of L.

With this in mind you get in your situation for n>>0

        \chi(F(x)G)(n) =  rk(F).rk(G).deg(L)/(d!) n^d + f(n)

where f(X) is a polynomial of degree at most d-1.

This can give you an upper bound of the dimensions for the sequence
(F(x)G)(n).

HTH.