You can NOT move money from your 1031 exchange into your IRA. A 1031 exchange is the sale of existing investment in real estate for another. Only if proceeds are reinvested in a like kind asset than no gains or losses are recognized deferring your capital gains. Basically, it is an unlimited tax break, as long as you roll the money from the investment into another real estate purchase within six months.
IRA has contribution limits per year.(This is the key here)(obviously it goes without saying if your 1031 exchange is within the allowable IRA contibution limit for the year you can do it, but that is common sense) It requires cash to fund it, or it allows in kind rollovers tax sheltered accounts(ie 401k or other IRA’s etc). You CAN purchase real estate inside your IRA once the money is in the account. The key is building your IRA up with cash/investments. 1031 exchanges in its very nature want a like kind exchange…real estate for real estate.

PS. Yes, alot of brokerage/banks will not allow for real estate purchases inside their IRA’s, because they are the custodian on the accounts and they don’t want the responsibility of a business they are not associated with(ie real estate). There are companies that do this, you just have to search for them. As far as you moving money into IRA from your 1031 exchange you are out of luck….but if you have money in your IRA, I am sure you can find a company that will allow for your real estate investments.