Papa John’s International Inc. says a consolidated antitrust lawsuit over a “no poach” clause in its franchise agreements should be dismissed or sent to arbitration.

The pizza chain filed a series of motions April 5 urging a Kentucky federal judge to toss three proposed class actions seeking to invalidate the provisions, which prohibit franchise operators from soliciting or hiring employees working for Papa John’s or another one of its franchises.

The suits, filed by one former and two current Papa John’s employees, claim the non-solicitation clauses are horizontal trade restraints that violate Section 1 of the Sherman Act on a...