In most cases yes. That is how other heterologous overexpression systems work. There may be a few cases in prokaryotes where regions downstream of the promoter are important for expression but not many if at all. In eukaryotes the regulation is much more complex and you may not always get the response you want, but in prokaryotes, especially with the standard promoters it should not be a problem.

I full agree with John. in most cases yes...