By default, the people who write the code own it, and that means the agency, not you. Paying the final invoice does not change that. Ownership moves to you only when a signed, written assignment says so. The accounts that hold your app are a separate question, and the best time to settle both is before the first line of code, not at handover.
The reason is how copyright works. The author of a work is its first owner, and an agency's engineers are the authors. Work made for hire sounds like a rescue. Under US law, though, a commissioned work qualifies only if it falls in one of nine listed categories and a signed agreement covers it. Custom software generally does not fit those categories. Without a writing, you hold a licence to use what was built. That is weaker than owning it. Rules differ by country and by contract, so have a lawyer read the real wording. This is the general picture, not legal advice.
A proper clause is short and uses the present tense. "The agency hereby assigns to the client all right, title and interest" transfers the work. "Agrees to assign" or "shall assign" only promises a transfer later, and a promise has to be enforced separately. It should name source code, documentation and anything derived from them. "Work product belongs to the client" is too loose to lean on, and "the client receives a perpetual licence" is a licence, not ownership.
Watch the payment trigger. Many contracts say the assignment takes effect on receipt of final payment, or when every invoice is paid in full. Read that as: the studio owns the code until the last invoice clears, and through any dispute along the way. Wanting protection against non-payment is fair, and a studio is entitled to it. The better shape ties the assignment to each milestone as it is paid, so the gap is small and known to both sides.
Ask about subcontractors too. An agency can only assign rights it holds. If a freelancer wrote part of your build, the studio needs that person's assignment first and has to pass it on to you. A short line in the contract that all contributors have signed one costs the studio nothing if it already does this.
Then there is background IP. Studios keep their own tools and frameworks, and open-source parts stay under their own licences. That's normal, and not worth fighting. What you want is a dated list of what is carved out, and a perpetual, irrevocable licence you can pass on to whoever maintains the app next. Be wary of a blanket carve-out for anything developed in connection with any engagement. Read loosely, that can swallow the custom work you paid for.
Now the part founders miss. Code is useless if someone else holds the doors. Your repository organisation, your Apple Developer and Google Play Console accounts, your cloud hosting, your domain and your payment and email services should all be registered to you or your company from day one, with the studio added as a collaborator. The Apple account controls your listing, your certificates and your ability to ship updates. The cloud account holds your servers, databases and user data, and losing it can take the app offline. Whoever holds the accounts has leverage over you, whatever the IP clause says.
Last, ask what a handover looks like. You want documentation good enough for another competent team to maintain the product, and no clause that strands your code or data if you leave. A studio sure of its work has no need for a lock-in.
Before you sign, put three questions to the studio. What do you keep, and where is it listed? Whose name is on each account? When exactly does the assignment take effect? Bring the answers to the first call. A studio that replies quickly, in writing, and without squirming is showing you how the project will run.