Quote from @Patricia Steiner:
A technicality first...the tenant always signs the lease first and then the landlord signs last to accept the lease. This is important because - if you haven't signed the lease nor have they, there is no implied or actual commitment to lease.
I would thank them for advising of their walk-through request and while it is not objectionable, you have moved forward with other interested applicants since they did not sign the lease when presented. As such, you feel an obligation to those other applicants in allowing them to finish their screenings and be considered as tenants.
There is no Fair Housing issue here as your decision not to hold a property for their continued inspection is not a criteria under the Act. By letting them know that you've moved forward with other applicants, you're letting them know that it is 'your ball, your court' - and they are not controlling the process. They may respond by wanting to sign the lease immediately, by accusing you of hiding something, or they may just go away. My recommendation would be to each time - regardless of their response - to thank them for following-up with you but as you've shared, you are moving foward with tenant selection and will hold their application for consideration.
I wouldn't be interested in this tenant; it's a power play on their part and one thing I know for sure:
What starts out crazy, ends crazier.
Bolt...
Very good advice. And yes, I always sign last. In this case, I drew up the lease and sent it for their signature, which they did not provide and informed me they wouldn't be signing it by the agreed move in date. So, none of us signed it.
If they are in any way demanding today, I was planning to tell them that we've decided to wait until we have a few more applications to consider. Unfortunately, as of yet, we have no other applications! But an empty house for a little while is worth more to me than a year of stress.