Looking for a word or phrase for a difficult transition that involves building momentum

Solution 1:

How about the idiom get into gear? From M-W:

get into gear: to start working or doing something in a more energetic and effective way; to cause (something) to start working or becoming successful

Your example:

Parents promise ice cream to help children get into gear so they can move on to the next thing on the agenda.

Get into gear suggests not only motion but momentum.

I also considered light a fire under someone but decided it was not only a poor fit but a dangerous way to get children to move more quickly. :-)

Solution 2:

"Parents promise ice cream to motivate their children."

Remarks: To do what the parents have asked. In the next sentence, you can describe what you hope that is. (Unless treats are repeated regularly, motivation will fade.)

Solution 3:

Get the ball rolling

Without rewriting too much, based on your phrase for incentivized motivation, "Parents can promise ice cream to get the ball rolling on the next task." Get the ball rolling and it will keep going.

It sounds like carrot ice cream, which can lead to stickiness.