Exhibit 10.4

Lock-Up Agreement

May [__], 2026

Dawson James Securities, Inc.

101 N. Federal Highway, Suite 600

Boca Raton, FL 33432

Ladies and Gentlemen:

The undersigned understands that Dawson James Securities, Inc. (the “Placement Agent”) has entered into a Placement Agency Agreement with Wellgistics Health, Inc., a Delaware corporation (the “Company”), which provides that Dawson James Securities, Inc. (the “Placement Agent”) shall serve as the exclusive placement agent for the Company, on a best efforts basis, in connection with the proposed private offering and placement (the “Offering”) by the Company of its securities, as further described in the securities purchase agreement (the “Securities Purchase Agreement”), dated as of May [__], 2026 , by and between the Company and the purchasers identified on the signature pages thereto (each, including its successors and assigns, a “Purchaser,” and collectively the “Purchasers”). Capitalized terms used and not otherwise defined herein that are defined in the Securities Purchase Agreement shall have the meanings given such terms in the Securities Purchase Agreement.

To induce the Placement Agent to continue its efforts in connection with the Offering and to induce the Purchasers to participate in the Offering, the undersigned hereby agrees that, without the prior written consent of the Placement Agent, during the period commencing on the Closing Date and ending on the earliest of (i) ninety (90) days after the Registration Statement is declared effective under the Securities Act, (ii) one hundred eighty (180) days after the Closing Date and (iii) such earlier date as the Placement Agent may agree in writing (the “Lock-Up Period”), the undersigned will not, and will not knowingly cause any entity controlled by the undersigned to, directly or indirectly, (1) offer, pledge, sell, contract to sell, grant, lend, or otherwise transfer or dispose of any shares of capital stock or any securities convertible into or exercisable or exchangeable for shares of capital stock, whether now owned or hereafter acquired by the undersigned or with respect to which the undersigned has or hereafter acquires the power of disposition (collectively, the “Lock-Up Securities”); (2) enter into any swap or other arrangement that transfers to another, in whole or in part, any of the economic consequences of ownership of the Lock-Up Securities; (3) establish or increase a put equivalent position or liquidate or decrease a call equivalent position within the meaning of Section 16 of the Securities Exchange Act of 1934, as amended, and the rules and regulations of the Commission promulgated thereunder with respect to any Common Stock owned directly by the undersigned or with respect to which the undersigned has beneficial ownership within the rules and regulations of the Commission, whether any such transaction described in clause (1) or (2) above is to be settled by delivery of Lock-Up Securities, in cash or otherwise; (4) file or cause to be filed any registration statement with the Commission relating to the offering of any shares of Common Stock or any securities convertible into or exercisable or exchangeable for shares of Common Stock, other than the Registration Statement, or make any demand for or exercise any right with respect to the registration of any Lock-Up Securities; or (5) publicly disclose the intention to make any offer, sale, pledge or disposition, or to enter into any transaction, swap, hedge or other arrangement relating to any Lock-Up Securities.

Notwithstanding the foregoing, the restrictions in this lock-up agreement shall not apply to: (a) transfers as a bona fide gift or charitable contribution; (b) transfers to any trust or other entity for the direct or indirect benefit of the undersigned or the immediate family of the undersigned; (c) transfers by will, intestacy or the laws of descent and distribution; (d) transfers to partners, members, stockholders or affiliates of the undersigned, if the undersigned is an entity; (e) distributions to limited partners, members, stockholders or other equity holders of the undersigned, if the undersigned is an entity; (f) transfers to the Company in connection with the exercise, vesting, settlement or tax withholding of any equity award, option, warrant or other convertible or exercisable security, provided that any shares received upon such exercise, vesting or settlement shall remain subject to this lock-up agreement; (g) transfers pursuant to a bona fide third-party tender offer, merger, consolidation, business combination or other similar transaction involving a change of control of the Company that has been approved by the Board of Directors of the Company; or (h) transfers with the prior written consent of the Placement Agent; provided, in each case, that the transferee agrees in writing to be bound by the terms of this lock-up agreement for the remainder of the Lock-Up Period to the extent such Lock-Up Period remains in effect.

The undersigned also agrees and consents to the entry of stop transfer instructions with the Company’s transfer agent and registrar against the transfer of the undersigned’s Lock-Up Securities except in compliance with this lock-up agreement.

Any release or waiver granted by the Placement Agent hereunder shall only be effective two (2) business days after the publication date of a press release announcing such release or waiver. The provisions of this paragraph will not apply if (a) the release or waiver is effected solely to permit a transfer of Lock-Up Securities not for consideration and (b) the transferee has agreed in writing to be bound by the same terms described in this lock-up agreement to the extent and for the duration that such terms remain in effect at the time of such transfer.

No provision in this lock-up agreement shall restrict or prohibit the exercise, exchange or conversion by the undersigned of any securities exercisable, exchangeable or convertible into Common Stock, including any exercise required pursuant to a mandatory exercise or call provision contained in any warrant; provided that any shares of Common Stock acquired upon such exercise, exchange or conversion shall remain subject to this lock-up agreement for the remainder of the Lock-Up Period unless otherwise permitted hereby. In addition, no provision herein shall restrict or prohibit the entry into, amendment or termination of a Rule 10b5-1 trading plan, provided that no sales of Lock-Up Securities may be made pursuant to such plan during the Lock-Up Period.

The undersigned understands that the Company, the Placement Agent and the Purchasers are relying upon this lock-up agreement in proceeding toward consummation of the Offering. The undersigned further understands that this lock-up agreement is irrevocable and shall be binding upon the undersigned’s heirs, legal representatives, successors and assigns.

This lock-up agreement is intended for the benefit of the parties hereto and their respective successors and permitted assigns (which do not include any investors in the offering referenced herein) and is not for the benefit of, nor may any provisions hereof be enforced by, any other person (including any investors in the offering referenced herein).

The undersigned understands that if the Securities Purchase Agreement (other than the provisions thereof which survive termination) is terminated prior to payment for and delivery of the securities to be sold thereunder, then this lock-up agreement shall be void and of no further force or effect.

Whether or not the Offering actually occurs depends on a number of factors, including market conditions. Any Offering will only be made pursuant to the Securities Purchase Agreement, the terms of which are subject to negotiation between the Company, the Placement Agent and the Purchasers.

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