Ils cherchent un meilleur équilibre entre leurs projets et leurs moments de repos.

French usually uses the simple present (ils cherchent) for both English they look for and they are looking for.

The progressive form être en train de + infinitive exists (ils sont en train de chercher), but it is:

• More specific and a bit heavier.
• Used when you really want to insist on the fact that the action is happening right now, at this very moment.

In a general sentence about life balance, French prefers the simple present:

• Ils cherchent un meilleur équilibre…
  = They are (in general, these days) looking for a better balance.

Using ils sont en train de chercher would sound like they are literally at this moment thinking about balance, which is a narrower meaning than the English present continuous here.

In French, chercher already includes the idea of for. It directly takes a direct object:

• chercher quelque chose = to look for something

So you say:

• Ils cherchent un meilleur équilibre…
  not ✗ Ils cherchent pour un meilleur équilibre…

Adding pour here is incorrect; it duplicates the function that chercher already has.

Both verbs are related, but they have different common uses.

Chercher:

• Everyday, neutral verb: to look for, to seek.
• Works for objects, people, abstract things:
  • Je cherche mes clés.
  • Nous cherchons une solution.

Rechercher:

• More formal, often used in written language or specific contexts.
• Can mean to search for, to seek, to research, to be in demand:
  • Cette qualité est très recherchée. (This quality is much sought after.)
  • Un laboratoire qui recherche un nouveau vaccin.

In your sentence:

• Ils cherchent un meilleur équilibre… is the normal, natural phrasing.
• Ils recherchent un meilleur équilibre… is possible, but sounds a bit more formal or “written”, as in a report or article.

Because équilibre is a masculine noun in French:

• un équilibre (a balance)
• le bon équilibre (the right balance)
• un meilleur équilibre (a better balance)

The adjective meilleur must agree in gender and number with the noun it modifies:

• Masculine singular: meilleur → un meilleur équilibre
• Feminine singular: meilleure → une meilleure situation
• Masculine plural: meilleurs
• Feminine plural: meilleures

So une meilleure équilibre is grammatically wrong because the article and adjective would be feminine while équilibre is masculine.

Meilleur and mieux both translate as better, but they’re used differently:

• Meilleur = better as an adjective (modifies a noun).

  • un meilleur équilibre (a better balance)
  • un meilleur travail (a better job)

• Mieux = better as an adverb (modifies a verb, adjective, or another adverb).

  • Ils dorment mieux. (They sleep better.)
  • C’est beaucoup mieux. (That’s much better.)

In un meilleur équilibre, better is describing the noun équilibre, so French needs the adjective form: meilleur, not mieux.
✗ un mieux équilibre is incorrect.

Un is an indefinite article: a / one.
Le is a definite article: the.

Here, un meilleur équilibre means “a better balance” in a general, non-specific sense:

• They don’t have a particular one exact balance identified.
• They just want some better kind of balance.

So:

• Ils cherchent un meilleur équilibre…
  = They are searching for a better balance (an improved state, not uniquely defined).

Le meilleur équilibre would mean the best balance, or the specific best possible balance, which is a different idea.

Entre means between and is followed by two (or more) things that you are positioning in relation to each other.

In the sentence:

• entre leurs projets et leurs moments de repos

you have the standard entre A et B structure:

• A = leurs projets
• B = leurs moments de repos

So the pattern is:

• un équilibre entre X et Y
  (a balance between X and Y)

You would not use parmi here. Parmi means among in the sense of within a group, not between two aspects that you’re trying to balance.

Leur / leurs must agree in number with the noun that follows, not with the owners.

• leur (no -s): before a singular noun
• leurs (with -s): before a plural noun

In the sentence:

• leurs projets: projets is plural → use leurs
• leurs moments de repos: moments is plural → use leurs

It does not matter whether ils refers to two people or ten; what matters is the noun:

• Ils aiment leur travail. (They love their job / work.) – travail is singular.
• Ils aiment leurs collègues. (They love their colleagues.) – collègues is plural.

So:

• leurs projets = their projects
• leur projet = their project (only one project)

You can grammatically drop the second leurs:

• entre leurs projets et leurs moments de repos (with repetition – more explicit)
• entre leurs projets et moments de repos (without repetition – still correct)

However:

• Repeating leurs (leurs projets et leurs moments de repos) is very clear and slightly more natural in careful or neutral style, especially in writing.
• Omitting the second leurs is more compact and a bit more typical in speech.

Both versions are acceptable; the original just makes the parallelism very clear.

Each option has a slightly different nuance:

• repos (without moments de)
  Can be more general or abstract:

  • Ils ont besoin de repos. (They need rest.)

• temps de repos
  Emphasizes time allocated for rest:

  • Ils n’ont pas assez de temps de repos. (They don’t have enough rest time.)

• moments de repos
  Suggests distinct, specific periods or little pockets of rest in their schedule:

  • leurs moments de repos = the particular moments in their lives/days when they actually rest.

In this sentence, leurs moments de repos evokes the idea of concrete restful moments that need to be balanced with their projects, rather than just an abstract concept of rest.

Yes, grammatically you could say:

• Ils cherchent un meilleur équilibre entre leur projet et leur moment de repos.

But the meaning changes:

• leurs projets (plural): suggests various projects, multiple things they are working on (work projects, personal projects, side projects, etc.).
• leur projet (singular): suggests one main project.

Similarly:

• leurs moments de repos: multiple moments of rest (evenings, weekends, breaks, vacations).
• leur moment de repos: one main or specific rest period.

The original plural form paints a picture of several projects and several rest periods that all need balancing.

Yes, that word order is grammatically correct:

• Entre leurs projets et leurs moments de repos, ils cherchent un meilleur équilibre.

Differences:

• Ils cherchent un meilleur équilibre entre leurs projets et leurs moments de repos.
  → Neutral, standard order, focus first on what they are looking for (un meilleur équilibre).

• Entre leurs projets et leurs moments de repos, ils cherchent un meilleur équilibre.
  → Puts initial emphasis on the two things being balanced (projects and rest).
  This is a bit more stylistic and is often used in written language for variety or emphasis.

Both are fine; the original is the most typical everyday phrasing.