Those killed belong to the Madhesi ethnic community, who are culturally close to Indians across the border.
Police told the BBC they were forced to open fire after they were attacked with guns and knives.
The new constitution will see Nepal divided into federal states, but the Madhesi and other groups fear they are being marginalised.
Of the five killed, officials said four were shot dead after protesters vandalised police posts and government offices in Birgunj, about 90km (55 miles) south-west of the capital, Kathmandu.
Another man was killed in the nearby town of Kalaiya.
The latest deaths bring the total killed in political violence in recent weeks to at least 20 people.
Nepal has been governed by an interim constitution since the monarch was abolished in 2008.
After years of wrangling, political parties only agreed on the draft this year after being given fresh impetus from April's devastating earthquake.
Its supporters hope the new constitution will boost economic development and cement peace with Maoist rebels who fought a decade-long insurgency.
But there is disagreement over the proposed borders, with minority groups saying they will not be granted sufficient rights.
Hindu groups have also been demonstrating, demanding that Nepal be declared a Hindu state - it became a secular republic after the monarchy was scrapped
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