The protein involved in "pushing" the tRNA and mRNA is the elongation factor EF-G. The protein involved in "transporting" the tRNA is the elongation factor EF-Tu.
Quoted from the Wikipedia articles:
When it binds to the ribosome A-site, EF-G causes the tRNA previously occupying that site to occupy an intermediate A/P position (bound to the A site of the small ribosomal subunit and to the P site of the large subunit), and the tRNA in the P site is shifted to a P/E hybrid state. EF-G hydrolysis of GTP causes a conformation change that forces the A/P tRNA to fully occupy the P site, the P/E tRNA to fully occupy the E site (and exit the ribosome complex), and the mRNA to shift three nucleotides down relative to the ribosome due to its association with these tRNA molecules. The GDP-bound EF-G molecule then dissociates from the complex, leaving another free A-site where the elongation cycle can start again
The tRNA anticodon domain associates with the mRNA codon domain in the ribosomal A site. If the codon-anticodon pairing is correct, EF-Tu hydrolyzes guanosine triphosphate (GTP) into guanosine diphosphate (GDP) and inorganic phosphate. This creates a conformational change in EF-Tu that causes EF-Tu to dissociate from the tRNA of the ternary complex (and therefore leave the ribosome). The aminoacyl-tRNA then fully enters the A site, where its amino acid is brought near the P site's polypeptide and the ribosome catalyzes the covalent transfer of the polypeptide onto the amino acid . The tRNA on the P site (without peptide) moves to the E site and is then released.