This functionality is available with Payments. You first have to filter to a specific client. After doing so, you will see checkboxes next to all invoices with a balance due. Select the invoices to be paid and click the Pay Invoices button.
This will generate a charge for each invoice. Keep in mind that if you have a single-use payment card to be applied to multiple invoices, enter the payment as a deposit, pay it with the card through Payments, and apply the deposit to the invoices.
Also, if your client creates a Payments account, they will already be filtered to just their client, so the checkboxes and "Pay Invoices" will be available to them when they log in.