Try to specialize some of the variables. Given enough specializations, one can then reconstruct kernel elements by interpolation.
Another method: try to use series expansions in terms of the variables and lift the order of such series expansions.
Another useful trick (when working with rational coefficients or coefficients in a number field) is working over finite fields since large determinant computations use up
huge amounts of memory. Use then either the Hensel lemma (working over $p-$adics) or combine
the information coming from different primes in order to lift the solution to $\mathbb Z$
or $\mathbb Q$.
Last ressort: If everything fails and if you are really interested in just one very specific
example, write a C-program for just your example using the best strategy you know and hope that it works.
There is probably no universally optimal way to do this, I guess you have to make advantage of any special features of your example.