This question has a ton of interesting nuance to explore and is the sort of topic that people could spend a career writing multiple books about. I’m going to explore just the broadest trends here – the elements that are shared in most cases – there are a lot of interesting and weird cases that I’d need access to my books to be comfortable talking about.

The short version is that if you were a noble going on Crusade you made sure to put your affairs in order and placed someone in charge of your lands/estate/responsibilities while you were away. It was very important to pick someone who was both capable and who you could trust. Your best options were usually either a close relative or the church.

Let’s talk about family first. You’ve actually chosen two interesting people as your examples (Richard I and Louis IX) because they both chose the same type of relative to act as regent in their absence: their mother. Richard I chose Eleanor of Aquitaine, a controversial but highly capable figure in European politics, to govern England in his absence while Louis IX had his mother, Blanche of Castile, act as regent for his first crusade. Blanche had acted as regent before, when Louis was a child, so her credentials were pretty well established by the time they were chosen. Mothers were generally a popular choice, they were seen as inherently trustworthy as they wouldn’t act against their own child’s interests and widowed women often filled the role of regent (usually for their underage children) so it fit into an expected social role.

Wives were a less popular choice of regent, but hardly unheard of. There could be some concern that a wife’s family could use her to undermine her husband’s family’s interests while he was away, which made them less popular, but also they would sometimes join their husbands on crusade or could be seen as too young and inexperienced to rule.

If you didn’t have a good candidate in your wife or mother, you could always ask your brother to take care of your lands for you. This was a little tricky, because siblings were known to undermine other siblings, but if your sibling was already your heir (i.e. if you didn’t have children) there was less of a need for them to do this. Also, there were decent odds on you not coming back from the Holy Land, so picking your brother made the inheritance smoother – this is especially true if your death was not immediately reported (or if you were captured and not ransomed). If your brother was a member of the clergy that was a great option – he probably wouldn’t undermine your inheritance and had the benefits of church protection, but this wasn’t always a practical option. If he lived far away, for example, he might not be able to keep your tenants in line.

If your son was of age to rule you could also pick him, but it would be more common for him to go on Crusade instead of you. Edward I is a famous example, having gone on crusade while still heir, and Richard I actually planned to go on crusade as a prince, but the death of Henry II changed that.

You’re second option was to turn to the church for protection. The Papacy was definitely concerned about crusaders’ lands being vulnerable while they were away, so early on they instituted protections on lands of nobles who took the cross. These included basic things like you couldn’t accumulate debts on your lands while you were away – limiting financial risk – but also included threatening interdict or excommunication on rulers who invaded or otherwise harmed the lands of a noble on crusade. These protections were helpful but didn’t always work in practice. A famous example is that of Duke Robert II of Normandy. He was the eldest son of William the Conqueror and inherited Normandy (but not England) from his father. He joined the First Crusade but to raise funds he had to mortgage his duchy to his brother, William II King of England. William died while Robert was on crusade and their younger brother, Henry, took the throne of England. This triggered a major conflict between the two when Robert returned that resulted in Henry capturing Robert and taking control of both England and Normandy. That’s a somewhat over simplistic description of a complicated event, but the general picture is one where Robert ended up much worse off for having been away from his lands while on Crusade. Another interesting case comes from Philip II of France, who returned early from the Third Crusade and used Richard I’s continued absence to undermine the English King’s rule over his French lands. Despite the church’s best wishes, being absent could cause problems for a ruler, but that’s not to say that the church’s rules were entirely ineffective.

You could also have a cleric rule over your lands directly. Louis VII chose Abbot Sugar of St. Denis to act as regent when he joined the Second Crusade. Sugar had been Louis’ teacher and was one of the premier clergy in the kingdom, so the choice made sense. One thing that was relatively common was for nobles to mortgage or sell part of their land to the church to raise funds for the Crusade. As part of this deal it could be arranged that the church (usually a local bishop or abbot) would take care of your lands in your absence. By offering them a financial stake in your lands you encouraged them to have both of your interests at heart in their stewardship.

It is also worth mentioning that these weren’t necessarily either/or situations. You could avail of some aspects of church protection while also placing a family member in charge. This would generally be most common if you had a young heir and you left them in the guardianship of a prominent member of the clergy (or sometimes joint guardianship between your wife and the church).

The most influential scholar in this area is probably still Jonathan Riley-Smith, who’s general history of the Crusades is still a great read.

Chris Tyerman’s How to Plan a Crusade offers lots of interesting detail across all aspects and areas of crusading.

Natasha Hodgson’s Women, Crusading and the Holy Land in Historical Narrative is about women and crusading generally, but offers tons of fascinating insight into regencies and the power of widows during this period.