One possible workflow:

1. Convert genomic annotations from GFF or GTF to BED format via gff2bed or gtf2bed, using grep to filter for genes
2. Add expression data to score column of BED-formatted gene annotations
3. Download or create chromosome extents for your genome of interest. For example, for hg19: hg19.extents.bed
4. Bin the extents by the desired 10kb increment with bedops --chop 10000
5. Pipe the bins to bedmap --echo --mean --fraction-map 0.51 to require that a gene overlaps a bin by 51% or more, in order to be mapped; the --mean option calculates the mean expression score value of any elements that map to the bin

Something greatly simplified as:

$ gff2bed < annotations.gff | grep -w 'gene' | cut -f1-4 - | paste - expression.txt > genes.bed5
$ bedops --chop 10000 hg19.extents.bed | bedmap --echo --mean --fraction-map 0.51 - genes.bed5 > answer.bed

A 51% threshold ensures that a gene that straddles two disjoint, adjacent bins will be mapped only to one or the other bin, depending on which gene subsegment has the greater overlap.