The phrase "push the process" is correct and usable in written English.
You can use it when you want to encourage someone to make progress on a task or activity. For example, you might say "Let's try to push the process and finish this project as soon as possible."
But there is only one truly persuasive reason to push the process forward now.
It is up to those in power in Thailand to push the process forward.
The international community, especially members of the security council, need to support the talks and take all measures to push the process forward.
Hoping to push the process forward, leaders from Latin America and the Caribbean have volunteered to aid in a reconciliation between the United States and Cuba.
Vice President Joseph Biden regularly called Iraqi officials for freewheeling conversations and visited Baghdad to try to push the process toward a conclusion.
The officials agreed to meet again in Geneva on June 25 to take stock of developments and try to push the process forward.
According to Mr. Felder, "the doorman blocked and pushed the process server and a police report was filed".
Earlier this year, a liberal budget pushed the process on, with tax cuts and renewed deregulation of the economy.
A trickle and then a flood of American records for the newly invented phonograph, and the presence of US troops in Europe, pushed the process on.
He left several staff members, including his top Middle East aide, behind to keep pushing the process, and was clearly trying to create momentum with his comments.
Pressure from campaigners has helped pushed the process this far; but such concerns suggest that it might now be time to make haste more slowly.
