我们先看词典例句。

《韦氏高阶》

She risked her life to save her children.

不定式为目的状语。目的一旦达到，risk的动作即告结束。目的状语动作后于谓语时间。

He risked all his money on starting his own business.

on sth/doing sth为方面状语。公司开始运作后，risk仍然在发生，不结束。即动名词的动作与谓语动作具有同时性。

《牛津》

 I can't risk anyone telling my parents about it.

动名词复合结构作宾语，anyone为动名词的逻辑主语。动名词的动作是谓语时间可能发生的动作。

你的句子1

At least the rulers of old had the decency to fight the war themselves and risk their lives doing so, instead of calmly sending men off to die while they sit in their office chair.

risk their lives (on) doing so. 有介词on, doing为动名词，介词短语为方面状语。省略介词on, 则分析为现在分词短语作时间状语或方面状语。doing与risk具有同时性。

你的句子2

International students share their views on why people would risk their lives to do these things.

不定式为目的状语。