What are modern full-time Catholic proselyting missions like? In reference to this question: What are full-time protestant proselyting missions like?

What are full-time Catholic proselyting missions like in contrast or comparison to LDS missions?
Generally Roman Catholics ('Catholics') do not use the term Proselytize due to the connotations associated with that term. I will therefore assume 'proselytize' in your question is synonymous with 'mission work'. (Pope Francis recently raised eyebrows by saying, in an interview, that “proselytism is solemn nonsense, it makes no sense.”)

"The Church’s missionary spirit is not about proselytizing, but the testimony of a life that illuminates the path, which brings hope and love. The Church—I  repeat once again—is not a relief organization, an enterprise or an NGO, but a community of people, animated by the Holy Spirit, who have lived and are living the wonder of the encounter with Jesus Christ and want to share this experience of deep joy, the message of salvation that the Lord gave us." [Message for World Mission Day, May 19, 2013].



Catholics are not required to do mission work; however, they are generally strongly encouraged to do volunteer work. The prevailing view in Catholicism is simply by the deeds done, people will be drawn to the church. That people will see the good and the love, and people will sua sponte explore the teachings of the Church. (This is one reason why Catholics do not preach 'fire and brimstone'.)

I would divide Catholic mission work into two categories: Lay missions and non-lay or ordained mission work. 

Much of the lay work is volunteer work that is organized by larger Catholic organizations. This idea is furthered by other US nonprofits like Catholic Charities USA, which spends nearly 90% of its income on charitable purposes, which are quite extensive.

The second group, ordained mission work, is a bit different. Generally, this mission work is done by religious rites, such as Missionaries of Charity, which was founded by Mother Teresa. Other rites emphasize bringing education to underprivileged areas, much as they did to the US in its early history.