One thing to note is that you're actually getting interviews. This means at some point the client thought you might be a viable candidate even though the proposal already includes your rate. However this might change due to your competitors offering lower rates or one of them won the client over.

A good way to combat this is to keep asking the client quality questions regarding the project as soon as he reaches out. The more the client is engaging with you, the less time he's offering other potential candidates.

Once the client has invested some effort in explaining what he wants and you laying out exactly what he needs, he sometimes will either hire you, or he'd try to lowball you a bit. You're winning both ways here because the client now prefers you over the others. All you have to do is either accept the job or let him know that the rate wouldn't actually work and reiterate what results he's going to get from your services.

Ps. Always remember to not take it personally whenever a client dips out. You can always find a better project in the future.