Before you compare payments, you are really answering two separate questions: how many miles will this car actually cover with kids in it...

Before you compare payments, you are really answering two separate questions: how many miles will this car actually cover with kids in it, and how long do you plan to keep it once you have it. Get those two answers first. Everything else, leasing versus buying, new versus used, follows from them rather than the other way around.
A car without children in the picture mostly does commuting and errands, and its mileage is fairly predictable. Add school runs, after-school activities spread across different buildings, weekend sport with equipment, grandparents who live an hour away, and the odd multi-day trip, and the mileage stops being a single commute multiplied by five. It becomes several separate loops stacked on top of each other, several times a week, for years.
This matters because it is the single biggest variable in whether leasing suits you. Lease agreements come with an annual mileage allowance built into the contract, and going over it costs you at the end, charged per mile for every mile past the limit. If you are underestimating your yearly total now, guessing low to keep the monthly payment down, you are setting up a bill that arrives later, when the car goes back.
Before you shop, do the arithmetic honestly. Take a normal week with the kids in their current activities and multiply the round trips by their actual distances, then multiply by roughly 45 weeks to allow for holidays and quieter stretches. Add anything seasonal: away sports fixtures, a summer visit to family further afield, a annual road trip. That total, not a number that felt round when you first heard it, is the mileage you should be comparing against any lease offer.
The monthly figure on a lease and the monthly figure on a loan are not measuring the same thing, and comparing them directly is where people go wrong.
A lease payment covers the use of the car for the lease term and nothing beyond that. It does not build you any ownership. A loan payment is buying you an asset, one you will still hold, and can sell or keep driving, after the payments stop.
To compare them properly, you need to look past the monthly number at:
None of this makes one option cheaper as a rule. It depends on how long you keep the car and how many miles you put on it, which is exactly why the mileage arithmetic above has to come first.
At the end of a lease term, you generally have three paths: hand the car back and walk away, hand it back and lease another one, or buy the car outright at a price set in the original contract.
Handing it back triggers an inspection. Normal wear is expected and allowed for, but scuffed alloys, torn upholstery, a cracked windscreen, or stains that will not come out can all turn into charges at return, on top of any mileage overage. With kids using the back seat daily, this is worth thinking about honestly rather than assuming it will be fine. Car seats scrape door frames. Snacks get dropped. A back seat that sees two children for three years shows it.
If you go over your mileage allowance, that charge is calculated per mile and added at return, which is why the estimate you did earlier matters more than it might seem to at signing.
Buying removes the mileage ceiling and the condition inspection entirely. The car is yours, and how it looks or how far it has travelled only affects what you could get if you chose to sell it, not a bill someone else presents to you.
The tradeoff is that you carry the maintenance and repair costs once any warranty runs out, and you carry the full risk of the car losing value faster than expected. A heavily used family car, covered in miles and years of daily wear, will typically be worth less at trade-in than a lightly used one of the same age. That is a cost too, just one that shows up as a smaller number when you eventually sell rather than as a bill in the post.
Leasing tends to suit a family with mileage that is genuinely moderate and predictable, who like driving a newer car with current safety features, and who would rather not think about a major repair bill landing without warning. It also suits someone whose situation might change in a few years, a move, a growing family needing a different size of car, where being free to walk away at the end has real value.
Buying tends to suit a family whose mileage is high, seasonal, or simply hard to predict from one year to the next, and who plan to keep a car well past the point where a lease term would end. It also suits anyone who wants to avoid an inspection at handback, or who would rather absorb wear and tear as their own problem than someone else's line item.
Neither of these is a verdict on which is smarter. They are a match between the arrangement and the shape of your actual driving.
Write down every recurring trip your family makes in a normal week, the school run, activities, and any regular longer drives, with real distances attached. Multiply the weekly total by 45 to account for a full year with some quieter weeks built in, then add any known seasonal trips. That single number, your honest annual mileage, is the first thing to check against any lease offer or the age and mileage of any used car you are considering. It will tell you more about which option fits than any monthly payment will.